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Please cite this article in press as: T. Duquesne, M. Winkel, Hereditary tree growth and Levy forests, Stochastic Processes and their Applications(2018), https://doi.org/10.1016/j.spa.2018.10.007.

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Stochastic Processes and their Applications ( ) –www.elsevier.com/locate/spa

Hereditary tree growth and Lévy forests

Thomas Duquesnea,∗, Matthias Winkelb

a Sorbonne Université, Faculté des Sciences Pierre et Marie Curie, Laboratoire de Probabilités Statistiques etModélisation, Boîte courrier 158, 4 Place Jussieu, 75252 Paris Cedex, France

b University of Oxford, Department of Statistics, 1 South Parks Road, Oxford OX1 3TG, UK

Received 21 September 2016; received in revised form 8 October 2018; accepted 15 October 2018Available online xxxx

Abstract

We introduce the notion of a hereditary property for rooted real trees and we also consider reductionof trees by a given hereditary property. Leaf-length erasure, also called trimming, is included as a specialcase of hereditary reduction. We only consider the metric structure of trees, and our framework is thespace T of pointed isometry classes of locally compact rooted real trees equipped with the Gromov–Hausdorff distance. We discuss general tightness criteria in T and limit theorems for growing familiesof trees. We apply these results to Galton–Watson trees with exponentially distributed edge lengths.This class is preserved by hereditary reduction. Then we consider families of such Galton–Watson treesthat are consistent under hereditary reduction and that we call growth processes. We prove that theassociated families of offspring distributions are completely characterised by the branching mechanism ofa continuous-state branching process. We also prove that such growth processes converge to Lévy forests.As a by-product of this convergence, we obtain a characterisation of the laws of Lévy forests in termsof leaf-length erasure and we obtain invariance principles for discrete Galton–Watson trees, including thesuper-critical cases.c⃝ 2018 Elsevier B.V. All rights reserved.

MSC: 60J80

Keywords: Real tree; Gromov–Hausdorff distance; Galton–Watson tree; Lévy tree; Leaf erasure; Limit theorems;Tightness; Invariance principle; Continuous-state branching process

This research was in part supported by EPSRC grant GR/T26368/01 and ANR GRAAL Project ANR-14 CE25-0014.∗ Corresponding author.

E-mail addresses: [email protected] (T. Duquesne), [email protected] (M. Winkel).

https://doi.org/10.1016/j.spa.2018.10.0070304-4149/ c⃝ 2018 Elsevier B.V. All rights reserved.

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2 T. Duquesne, M. Winkel / Stochastic Processes and their Applications ( ) –

1. Introduction

The purpose of this paper is to introduce a general class of tree reduction and tree growthprocedures that we call “hereditary”, and to apply this notion in the study of limit theoremsfor Galton–Watson forests, both as almost sure (increasing) limits and as distributional limitsin the form of an invariance principle. Here we view trees as certain metric spaces called realtrees and we are using and developing a framework initiated by Aldous [3,4] and Evans, Pitmanand Winter [15] who first considered in our probabilistic context the space of compact realtrees equipped with the Gromov–Hausdorff distance. The large class of growth processes ofGalton–Watson forests that we consider in this framework contains the important example offorests consistent under leaf-length erasure (see Neveu [36], and Le Jan [34] in the context ofsuper-processes), and it is also closely related to two specific models considered by Geiger andKauffmann [18] and by the present authors in [13]. We prove that in some sense, any way ofgrowing Galton–Watson trees yields, in the limit, Levy trees, which are continuum random treesintroduced by Le Gall and Le Jan [33] that have been further studied in [11] and also by Abraham,Delmas [1] and Weill [42].

Let us briefly review in this introduction the main results of the paper. First we recall a fewdefinitions on real trees and the space of trees we consider. A real tree is a path-connectedmetric space (T, d) with the following property: any two points σ, σ ′ ∈ T are connectedby a unique injective path denoted by [[σ, σ ′]], which furthermore is isometric to the interval[0, d(σ, σ ′)] of the real line. Informally, real trees are obtained by gluing together, withoutcreating loops, intervals of R equipped with the usual metric. However, note that real trees mayhave a complicated local structure, like Aldous’s (Brownian) Continuum Random Tree, whichis a compact real tree whose set of leaves is uncountable and dense. In each real tree (T, d), wedistinguish a point ρ ∈ T that is viewed as the root. So we speak of (T, d, ρ) as a rooted realtree.

We shall focus on complete locally compact rooted real trees (CLCR real trees for short). Wethen say that two CLCR real trees are equivalent if there exists a root-preserving isometry fromone tree onto the other. We simply denote by T the pointed isometry class of a given CLCR realtree (T, d, ρ). We denote by T the set of pointed isometry classes of CLCR real trees. We equipT with the pointed Gromov–Hausdorff distance denoted by δ (see Section 2.1 for a definition).Then (T, δ) is a Polish space. This result is due to Gromov [22] for compact metric spaces and toEvans, Pitman and Winter [15] for compact real trees (see also [13] for the standard adaptation topointed CLCR real trees). The main notions and results of this paper (reduction under hereditaryproperties, tree growth processes, limit theorems and invariance principles) take place in thespace (T, δ).

Let us briefly explain the notion of a hereditary property in this context. Let (T, d, ρ) be aCLCR real tree. Then we define for every σ ∈ T the subtree above σ as θσT = σ ′ ∈ T : σ ∈[[ρ, σ ′]]. Note that (θσT, d, σ ) is also a CLCR real tree. We denote by θσT its pointed isometryclass in T. A hereditary property is a Borel subset A ⊂ T such that for every CLCR real tree(T, d, ρ) and for every σ ∈ T , if θσT ∈ A, then T ∈ A. In order to rephrase this definitioninformally, let us view T as a continuum of individuals whose progenitor is the root ρ: if anindividual σ ∈ T “has the hereditary property A”, namely if θσT ∈ A, then the progenitor ρalso “has the property A”; implicitly, a hereditary property may be lost on the ancestral lineagebetween the progenitor and an individual, and an individual can only inherit a hereditary propertyif all his ancestors had it.

We then define the A-reduced subtree of T as RA(T ) where

RA(T ) is the closure in T of the subset ρ ∪ σ ∈ T : θσT ∈ A.


T. Duquesne, M. Winkel / Stochastic Processes and their Applications ( ) – 3

Then, (RA(T ), d, ρ) is a CLCR real tree and its pointed isometry class only depends on theisometry class of T . Hence, there is an induced function from T to T that we simply denoteby RA. Hereditary properties can be composed in the following sense: let A, A′ ⊂ T be twohereditary properties, we then set A′ A = T ∈ T : RA(T ) ∈ A′ and Lemma 2.13 asserts thatA′ A is hereditary and moreover RA′A = RA′ RA.

The most important example of hereditary reduction is the leaf-length erasure (also calledtrimming) that is defined as follows. For any CLCR real tree (T, d, ρ), denote by Γ (T ) =supσ∈T d(ρ, σ ) its total height (that is possibly infinite). Since it only depends on T , it induces afunction on T, that is also denoted by Γ and that is δ-continuous. Then for any h ∈ [0,∞), weset Ah = T ∈ T : Γ (T ) ≥ h, which is clearly hereditary. We shall simply write RAh as Rh andrefer to Rh as the h-leaf erasure. Note that for any h, h′ ∈ [0,∞), Ah′ Ah = Ah+h′ and thus,Rh′ Rh = Rh′+h .

Leaf-length erasure was first considered by Kesten [28] for discrete trees. Then it was studiedby Neveu [36], and by Neveu and Pitman [37] to approximate the Brownian tree (see alsoLe Gall [32]); later Le Jan [34] used it to construct superprocesses with a stable branchingmechanism. In the context of compact real trees, leaf-length erasure was more systematicallyused by Evans, Pitman and Winter [15]. They proved in particular that Rh is δ-continuous.

One important fact to note is that for any CLCR real tree T , Rh(T ) is a discrete real tree, thatis Rh(T ) has a discrete branching structure: the set of branch points has no accumulation pointsand all branch points have finite degree. For every a ∈ [0,∞) and every h ∈ (0,∞), we set

Z (h)a (T ) = # σ ∈ Rh(T ) : d(ρ, σ ) = a .

Since Rh(T ) is a discrete real tree, a ↦→ Z (h)a (T ) is an N-valued function that is left-continuous

with right limits. We call the process a ↦→ Z (h)a (T ) the h-erased profile. Since Z (h)

a (T ) onlydepends on the pointed isometry class of T , it induces a function on T that is denoted in thesame way.

As proved in Lemma 3.1, the erased profiles allow to control the covering numbers of ballsof CLCR real trees, which is the key argument in the following general tightness criterion in(T, δ): let (T j ) j∈J be a family of T-valued random trees. Their laws are δ-tight if and only iffor every fixed h, a ∈ (0,∞), the laws of the N-valued random variables (Z (h)

a (T j )) j∈J are tight.This is Theorem 3.3, which complements existing tightness criteria (e.g. [22,16]). This tightnesscriterion is used to obtain, among other results, invariance principles for Galton–Watson trees(see Theorem 3.25 in Section 3.4). More generally, it is well-adapted to any model of randomtrees that allows a certain control on the erased profiles.

In this paper we consider tree-valued processes that grow. One simple way to understand thegrowth is to view the trees in a certain ambient metric space and to say that the trees grow withrespect to the inclusion partial order. However, we only consider the metric structure and wewant to consider neither the ambient metric spaces that allow such constructions nor the detailsconcerning the many ways the tree-growth process can be embedded in a given ambient space.This is why we introduce the following intrinsic definition of growth: let T , T ′ ∈ T; we say thatT can be “embedded” in T ′, which is denoted by T ⪯ T ′, if we can find representatives (T, d, ρ)and (T ′, d ′, ρ ′) of T and T ′, and an isometrical embedding f : T → T ′ such that f (ρ) = ρ ′.

Note that ⪯ is a partial order on T. Moreover, for any hereditary property A, if T ⪯ T ′

then RA(T ) ⪯ RA(T ′) and in particular for any h > 0, Rh(T ) ⪯ Rh(T ′), which implies forany a ∈ [0,∞), Z (h)

a (T ) ≤ Z (h)a (T ′). This yields the following convergence criterion stated

in Theorem 3.9: let (Tn)n∈N be a T-valued sequence of random trees such that for all n ∈ N,Tn ⪯ Tn+1 almost surely; furthermore assume that for every fixed a, h ∈ (0,∞), the family of



laws of the N-valued random variables Z (h)a (Tn), n ∈ N, is tight; then, there exists a random tree

T in T such that limn→∞δ(Tn, T ) = 0 a.s.This result is used to prove that, when convergent, any growing family of Galton–Watson

forests tends to either a Galton–Watson forest or to a Levy forest. Before explaining this result,let us informally discuss the model of Galton–Watson real forests that are considered here (theyare introduced precisely in Section 2.3). Their laws are characterised by the following threeparameters: the offspring distribution ξ = (ξ (k))k∈N, the lifetime parameter c ∈ (0,∞), and theinitial distribution µ = (µ(k))k∈N. We view a Galton–Watson real forest with parameters ξ , cand µ (a GW(ξ, c;µ)-real forest for short) as the forest of genealogical trees of a population thatevolves as follows: at generation 0, the population has N independent progenitors, where N haslaw µ; the lifetimes of individuals are independent and exponentially distributed with mean 1/c;when they die, individuals independently give birth to a random number of children distributedaccording to ξ . Here, we shall assume that ξ is proper and non-trivial, namely that ξ (1) = 0and ξ (0) < 1. Moreover we also assume that the associated continuous-time N-valued branchingprocess is conservative (see (27) in Section 2.3 for more details).

Theorems 2.19 and 2.18 assert that the class of GW-laws is preserved by hereditary reduction.More precisely, let F be a GW(ξ, c;µ)-real forest, let A ⊂ T be a hereditary property and denoteby α the probability that the A-reduced tree of a single GW(ξ, c)-real tree is just a point. Assumethat α ∈ (0, 1). Then RA(F) is a GW(ξ ′, c′;µ′)-real forest, where (ξ ′, c′;µ′) is given in terms ofα and (ξ, c;µ) as follows: if we denote by ϕν the generating function of a probability measure νon N, then

ϕξ ′ (r ) = r +ϕξ (α + (1− α)r )− α − (1− α)r

(1− α)(1− ϕ′ξ (α)),

c′ = (1− ϕ′ξ (α)) c and ϕµ′ (r ) = ϕµ(α + (1− α)r ).(1)

This important property of hereditary reduction naturally leads us to consider families of GW-real forests that are consistent under hereditary reduction. Namely, we call a family (Fλ)λ∈[0,∞)

of Galton–Watson real forests a growth process if for all λ′ ≥ λ there exist hereditary Aλ,λ′ ⊂ T,such that almost surely

∀λ′, λ ∈ [0,∞) such that λ′ ≥ λ, Fλ = RAλ,λ′ (Fλ′ ) .

Let us say that Fλ is a GW(ξλ, cλ, µλ)-real forest and assume that µλ tends to the Dirac massat infinity, as λ → ∞. This assumption implies that each offspring distribution ξλ appears asa reduced law as in (1) for all α sufficiently close to 1 (such offspring distributions are calledinfinitely extensible [13]). Then Theorem 3.16 shows that they are quite specific. Namely, thelaws (ξλ, cλ, µλ) are entirely governed by a triplet (ψ, β, ϱ) defined as follows.

• ψ : [0,∞) → R is the branching mechanism of a continuous-state branching process.Namely, ψ is the Laplace exponent of a spectrally positive Levy process and it is thereforeof the Levy–Khintchine form

ψ(λ) = aλ+ 12 bλ2

+

∫(0,∞)

(e−λx− 1+ λx1x<1

)π (dx) ,

where a ∈ R, b ∈ [0,∞) and π is a Borel measure on (0,∞) such that∫

(0,∞)(1 ∧x2)π (dx) < ∞. Moreover, ψ has to satisfy two additional conditions: it takes positivevalues eventually and

∫0+ dr/|ψ(r )| = ∞ (the latter assumption is equivalent to assuming

that the continuous-state branching process governed by ψ is conservative).



• β : [0,∞)→ [0,∞) is non-decreasing and such that limλ→∞β(λ) = ∞.• ϱ is a probability measure on [0,∞) that is distinct from the Dirac mass at zero.

Then, for any λ ∈ [0,∞), (ξλ, cλ, µλ) is derived from (ψ, β, ϱ) as follows:

ϕξλ (r ) = r +ψ((1− r )β(λ))β(λ)ψ ′(β(λ))

, cλ = ψ ′(β(λ)) and ϕµλ (r ) =∫

(0,∞)e−yβ(λ) ϱ(dy).

These offspring distributions already appeared in the more specific context of [11,13]. Ofparticular interest are cases where the offspring distributions ξλ are all equal to a certain ξ notdepending on λ ≥ 0. We easily see that this exactly corresponds to the stable cases whereψ(λ) = λγ , for a certain γ ∈ (1, 2] and thus ϕξ (r ) = r + 1

γ(1 − r )γ , cf. the discussion

after Definition 3.11. We call these laws the γ -stable offspring distributions. The Browniancase corresponds to the critical binary offspring distribution (γ = 2). They appear in previouswork [34,36] (and in a slightly different form in [11,41,26]).

Observe that λ ↦→ Γ (Fλ) is non-decreasing almost surely, since the growth process is ⪯-non-decreasing. Then limλ→∞Γ (Fλ) exists in [0,∞] and standard branching process argumentsgive that P(limλ→∞Γ (Fλ) < ∞) > 0 iff ψ satisfies

∫∞dr/ψ(r ) < ∞. This is a necessary

and sufficient condition for the growth process to converge in (T, δ), almost surely. Namely,Theorem 3.19 asserts that there exists a random CLCR real tree F such that

limλ→∞

δ(Fλ, F

)= 0 almost surely.

The limiting tree F is a ψ-Levy forest. Moreover, the branching processes associated to Fλ,λ ≥ 0, also converge almost surely. Namely, for any a ∈ [0,∞), we get

1β(λ)

# σ ∈ Fλ : d(ρ, σ ) = a −−−−→λ→∞

Za almost surely,

where (Za)a∈[0,∞) is a continuous-state branching process with branching mechanism ψ andwhose initial value Z0 has law ϱ (see Section 3.3 for more details). As an application of theseresults, Theorem 3.20 provides a nice characterisation of Levy forests via leaf-length erasure.This result is used in the proof of the invariance principles for discrete Galton–Watson trees thatfollow in Section 3.4. These limit theorems are general and they notably include the super-criticalcases. Invariance principles for critical and sub-critical trees were obtained in [11] using differentarguments and a different formalism.

This paper is organised as follows. In Section 2.1, we recall basic definitions concerning realtrees and the Gromov–Hausdorff metric, and we discuss various operations on real trees. Thetechnical details concerning the measurability of such operations are postponed to an appendix.Hereditary reduction is studied in Section 2.2, and Section 2.3 discusses an intrinsic definitionof Galton–Watson real trees. Section 3.1 is devoted to the statements and proof of tightnessand convergence criteria for real trees. In Section 3.2, we define and study growth processesof Galton–Watson forests, whose limits are discussed in Section 3.3. Section 3.4 is devoted toinvariance principles for rescaled discrete Galton–Watson trees with unit edge lengths.

2. Real trees, hereditary properties and Galton–Watson trees

2.1. Real trees

Real trees form a class of loop-free length spaces, which turn out to be limits of many discretetrees. More precisely, we say that a metric space (T, d) is a real tree if it satisfies the followingconditions:



• for all σ, σ ′ ∈ T , there exists a unique isometry fσ,σ ′ : [0, d(σ, σ ′)] → T such thatfσ,σ ′ (0) = σ and fσ,σ ′ (d(σ, σ ′)) = σ ′; we set [[σ, σ ′]] := fσ,σ ′ ([0, d(σ, σ ′)]), which is thegeodesic joining σ to σ ′;• if q : [0, 1]→ T is continuous injective, we have q([0, 1]) = [[q(0), q(1)]].

Let ρ be a distinguished point of T , which is viewed as the root of T . Then (T, d, ρ) iscalled a rooted real tree. We also denote by ]]σ, σ ′]], [[σ, σ ′[[ and ]]σ, σ ′[[ the respective imagesof (0, d(σ, σ ′)], [0, d(σ, σ ′)) and (0, d(σ, σ ′)) under fσ,σ ′ . We view the tree T as the family treeof a population whose progenitor is ρ. For any σ, σ ′ ∈ T , their most recent common ancestor isthen the unique point denoted by σ ∧ σ ′ such that [[ρ, σ ]] ∩ [[ρ, σ ′]] = [[ρ, σ ∧ σ ′]]. Thus,

∀σ, σ ′ ∈ T , d(ρ, σ ∧ σ ′) =12

(d(ρ, σ )+ d(ρ, σ ′)− d(σ, σ ′)

). (2)

There is a nice characterisation of real trees in terms of the four points condition: a connectedmetric space (X, d) is a real tree iff for all σ1, σ2, σ3, σ4 ∈ X ,

d(σ1, σ2)+ d(σ3, σ4) ≤ (d(σ1, σ3)+ d(σ2, σ4)) ∨ (d(σ3, σ2)+ d(σ1, σ4)). (3)

We refer to [8–10] for general results concerning real trees, to [38,39] for applications of realtrees to group theory and to [11,12,14–16] and [23] for probabilistic use of real trees.

Gromov–Hausdorff distance on the space of complete locally compact real trees. We say thattwo pointed metric spaces (X1, d1, ρ1) and (X2, d2, ρ2) are equivalent iff there exists a pointedisometry, i.e. an isometry f from X1 onto X2 such that f (ρ1) = ρ2. The Gromov–Hausdorffdistance of two pointed compact metric spaces (X1, d1, ρ1) and (X2, d2, ρ2) is given by thefollowing.

δcpct(X1, X2) = inf dHaus( f1(X1), f2(X2)) ∨ d( f1(ρ1), f2(ρ2)) . (4)

Here the infimum is taken over all ( f1, f2, (E, d)), where (E, d) is a metric space, wherefi : X i → E , i = 1, 2, are isometric embeddings and where dHaus stands for the Hausdorffdistance on the set of compact subsets of E . Observe that δcpct only depends on the isometryclasses of the X i . It induces a metric on the set of isometry classes of all pointed compact metricspaces (see [22]).

We then denote by Tcpct the set of pointed isometry classes of compact rooted real trees.Evans, Pitman and Winter [15] showed that Tcpct equipped with the Gromov–Hausdorff distanceδcpct is a complete and separable metric space.

We then denote by T the set of pointed isometry classes of complete locally compact rootedreal trees. The Gromov–Hausdorff distance is extended to T in the following standard way. Let(T1, d1, ρ1) and (T2, d2, ρ2) be two complete locally compact rooted real trees. Recall that theHopf–Rinow theorem (see [22, Chapter 1]) implies that the closed balls of T1 and T2 are compactsets (note that this entails that T1 and T2 are separable). We then set

δ(T1, T2) =∑k≥1

2−kδcpct(BT1 (ρ1, k), BT2 (ρ2, k)),

where for any i ∈ 1, 2, BTi (ρi , k) stands for the closed (compact) ball with centre ρi and radiusk in the locally compact rooted real tree (Ti , di , ρi ). Clearly, δ only depends on the isometryclasses of T1 and T2. It defines a metric on T and (T, δ) is Polish (see [13, Proposition 3.4]).

Notation and convention. We shorten Complete Locally Compact Rooted real tree to CLCR realtree. Let (T, d, ρ) be a CLCR real tree. We denote by T ∈ T its pointed isometry class. We shalldenote by Υ the pointed isometry class of a point tree (ρ, d, ρ).



Approximation by discrete real trees. Let (T, d, ρ) be a rooted real tree. For all σ ∈ T we denoteby n(σ, T ) the degree of σ , namely the (possibly infinite) number of connected components ofT \ σ . We also introduce the following

Lf(T ) = s ∈ T \ ρ : n(σ, T ) = 1 and Br(T ) = s ∈ T \ ρ : n(σ, T ) ≥ 3

that are respectively the set of the leaves of T and the set of branch points of T . Note that theroot is neither considered as a leaf nor as a branch point. Recall from [13] that the set of branchpoints of a CLCR real tree is at most countable.

Definition 2.1. A rooted real tree (T, d, ρ) is called a discrete real tree if it is complete and if

∀R ∈ (0,∞) , n(ρ, T )+∑

n(σ, T ) <∞ , (5)

where the sum is over all the branch points σ ∈ Br(T ) such that d(ρ, σ ) ≤ R.

An equivalent definition is the following: a complete real tree (T, d, ρ) is a discrete real treeiff

(a) the degree of branch points is bounded in every ball of finite radius;(b) every ball contains a finite number of branch points and a finite number of leaves.

Indeed, the only non-trivial point to check is that (5) implies that every ball contains a finitenumber of leaves. We argue by contradiction: fix R ∈ (0,∞) and suppose that there exists asequence (σn)n∈N of distinct leaves such that d(ρ, σn) ≤ R. Let B be the set of points γ such thatγ = σi ∧ σ j for infinitely many pairs of integers i < j . Clearly, B ⊆ Br(T ) ∩ BT (ρ, R) and (5)implies that B is finite and non-empty. Let β ∈ B be such that d(ρ, β) = maxγ∈Bd(ρ, γ ). By(5), β has finite degree (say n + 1); let T1, . . . , Tn be the open connected components of T \ βthat do not contain the root. One of these connected components Ti has to contain infinitely manyterms of the sequence (σn)n∈N. Then there is β ′ ∈ Ti ∩ B such that d(ρ, β ′) > d(ρ, β), which isnot possible.

Let (T, d, ρ) be a discrete real tree. Denote the connected components of T \ (Br(T )∪ρ) byCi , i ∈ I . The components Ci , i ∈ I , are called the edges of T . They all are isometric to intervalsof the real line. Note that their endpoints are leaves, branch points or the root. By (a) and (b), forall R ∈ (0,∞), only finitely many of such edges have points at distance less than R from theroot. This implies that the closed ball with centre ρ and radius R is compact. Thus, (T, d, ρ) islocally compact. Let us denote by T∗ the set of pointed isometry classes of the discrete real trees.Then, we get

T∗ ⊂ T .

We next introduce leaf-length erasure (also called trimming) that is used throughout the paperas a technical tool. Moreover, it is one of the most useful examples of hereditary reduction, aswe discuss later in Section 2.2. Let h ∈ (0,∞) and let (T, d, ρ) be a CLCR real tree. We set

Rh(T ) = ρ ∪ σ ∈ T : ∃σ ′ ∈ T such that σ ∈ [[ρ, σ ′]] and d(σ, σ ′) ≥ h. (6)

Clearly, Rh(T ) is path-connected and it is easy to prove that it is a closed subset of T . Then, thefour points condition implies that (Rh(T ), d, ρ) is a CLCR real tree. We call it the h-leaf-lengtherased tree associated with T . Note that its isometry class only depends on that of T so Rh canbe defined from T to T.



Leaf-length erasure was first introduced by Kesten [28] for discrete trees and further studiedand applied by many others [36,37,32,34,15]. In the following lemma we sum up the variousproperties of the leaf-length erasure operator Rh that we shall use in this paper; it is astraightforward extension of the same result for compact trees that is due to Evans, Pitman andWinter [15, Lemma 2.6].

Lemma 2.2.

(i) For every h ∈ (0,∞) and every CLCR real tree (T, d, ρ), (Rh(T ), d, ρ) is a discrete realtree.

(ii) For every h ∈ [0,∞), Rh is δ-continuous.(iii) Rh+h′ = Rh Rh′ , h, h′ ∈ [0,∞).(iv) For every h ∈ [0,∞) and for every CLCR real tree (T, d, ρ), we have δ(Rh(T ), T ) ≤ h.

Indeed, since T is locally compact, we immediately get that n(σ, Rh(T )) < ∞, for everyσ ∈ Rh(T ). To show (i), it remains to show that Br(Rh(T )) has no limit points. Suppose that thereexists a sequence (σn)n∈N of distinct branching points of Rh(T ) converging to σ ∈ Rh(T ). Thenit is easy to see that the closed ball BT (σ, 2h) is not locally compact, which is a contradiction.Points (iii) and (iv) are easy consequences of [15, Lemma 2.6]. Let us briefly explain (ii): forevery CLCR real tree (T, d, ρ) and for every r > 0, we set T (h, r ) = Rh(BT (ρ, r + 2h)). Thenobserve that BT (h,r )(ρ, r ) = Rh(T ) ∩ BT (ρ, r ). This entails (ii) by [15, Lemma 2.6(ii)] , whichasserts that Rh is δcpct-continuous on Tcpct.

We introduce functions of real trees such as the total height, the profile and various proceduresthat allow to split or to paste trees. Continuity or measurability of such functions is quiteexpected, however some of the proofs are technical. Thus, to ease the reading, we postponethem to Appendix A.

The total height. Let (T, d, ρ) be a CLCR real tree. The total height of T is given by

Γ (T ) = supσ∈T

d(ρ, σ ) ∈ [0,∞] .

Note that Γ (T ) only depends on the pointed isometry class T of (T, d, ρ): it induces a functionon T that is denoted in the same way. It is easy to check that Γ is δ-continuous and note that

∀h ∈ [0,∞) , Γ Rh = (Γ − h)+. (7)

The h-erased profile. For every a ∈ [0,∞) and every h ∈ (0,∞), we set

Z (h)a (T ) = # σ ∈ Rh(T ) : d(ρ, σ ) = a . (8)

Since Rh(T ) is a discrete real tree, it is easy to check that a ↦→ Z (h)a (T ) is an N-valued function

that is left-continuous with right limits. We call the process a ↦→ Z (h)a (T ) the h-erased profile.

We shall denote by Z (h)a+(T ) its right limit at a. Note that h ↦→ Z (h)

a (T ) is non-increasing. SinceZ (h)

a (T ) only depends on the pointed isometry class of T , it induces a function on T that isdenoted in the same way.

Splitting measures. Let us recall that a point measure on T is a measure of the form∑

i∈I δTi,

where Ti , i ∈ I , is any finite (possibly empty) or countably infinite collection of not necessarilydistinct elements Ti ∈ T. We introduce the following set of point measures

M (T) =

∑i∈I

δTipoint measure on T : ∀h ∈ (0,∞), #i ∈ I : Γ (Ti ) > h <∞

.



Note that M (T) contains the null measure (for which I is taken empty). We use the followingstandard notation on point measures: for every M =

∑i∈I δTi

in M (T) and for every functionG : T → [0,∞), we set ⟨M,G⟩ =

∑i∈I G(Ti ). We shall also denote by ⟨M⟩ the total mass of

M : we then check that ⟨M⟩ = ⟨M, 1T⟩ = #I .We equip M (T) with the sigma-field GM (T) generated by the functions M ∈ M (T) ↦→

⟨M,G⟩, for G : T→ [0,∞) Borel-measurable.We also define the trees above level a as follows. Let (T, d, ρ) be a CLCR real tree. Denote

by T i (a), i ∈ I (a), the connected components of the (possibly empty) open set T \ BT (ρ, a);for every i ∈ I (a), denote by Ti (a) the closure of T o

i (a) in T and denote by σi (a) the uniquevertex such that Ti (a) = T i (a) ∪ σi (a). Observe that d(ρ, σi (a)) = a. Clearly, the trees(Ti (a), d, σi (a)), i ∈ I (a), are CLCR real trees whose pointed isometry classes are denotedby Ti (a), i ∈ I (a). We then set

Ma(T ) =∑

i∈I (a)

δTi (a) ∈M (T) and Z+a (T ) = ⟨Ma(T )⟩ = #I (a) ∈ N ∪ ∞. (9)

The fact that Ma(T ) ∈ M (T) is an easy consequence of the local compactness of T . Themeasure Ma(T ) is called the splitting measure of T at level a and Z+a (T ) is called the rightprofile of T . Note that these functions only depend on T : they induce functions on T that aredenoted in the same way. It is easy to check that for every a ∈ [0,∞) and every h ∈ (0,∞),

Z (h)a+ = Z+a Rh and Z+a = lim

h→0+Z (h)

a+. (10)

Grafting real trees. Let us explain how to graft real trees on the vertices of another real tree.Let (T, d, ρ) be a rooted real tree, let (Ti , di , ρi ), i ∈ I , be a family of real trees and let σi ∈ T ,i ∈ I , be a collection of vertices of T . We then set

T ′ = T ⊔⨆i∈I

Ti \ ρi ,

where ⊔ stands for the disjoint union. We next define a metric d ′ on T ′ × T ′ as follows: d ′

coincides with d on T×T ; assume that σ ∈ Ti\ρi ; if σ ′ ∈ T , then we set d ′(σ, σ ′) = di (σ, ρi )+d(σi , σ

′); if σ ′ ∈ T j \ ρ j with j = i , then we set d ′(σ, σ ′) = di (σ, ρi )+ d(σi , σ j )+ d j (σ ′, ρ j );if σ ′ ∈ Ti \ ρi , then we set d ′(σ, σ ′) = di (σ, σ ′). It is easy to prove that (T ′, d ′, ρ ′ := ρ) is arooted real tree and we use the notation

(T ′, d ′, ρ ′) = T ⊛i∈I (σi , Ti ). (11)

We shall extensively use the following special case: we assume that T reduces to its root T = ρ:in this case σi = ρ and we use the specific notation

⊛i∈I Ti := ρ⊛i∈I (ρ, Ti ) ,

with the convention that ⊛i∈I Ti = ρ if I is empty. In words, ⊛i∈I Ti is a tree obtained bypasting the trees Ti , i ∈ I , at their roots.

We now assume that for every i ∈ I , (Ti , di , ρi ) is a CLCR real tree and we assume that forevery h ∈ (0,∞), #i ∈ I : Γ (Ti ) > h <∞. This implies that i ∈ I : Ti = ρi is countable.It is easy to check first that ⊛i∈I Ti is a CLCR real tree. Note that its pointed isometry class onlydepends on the pointed isometry classes of Ti , i ∈ I . We denote by ⊛i∈I Ti the pointed isometryclass of ⊛i∈I Ti .



This induces a function Paste : M (T) → T, that is defined as follows: for every M =∑i∈I δTi

∈M (T), we set

Paste (M) = ⊛i∈I Ti , (12)

with the convention that Paste (0) = Υ for the null measure 0 ∈M (T). It is easy to check that

∀T ∈ T , Paste (M0(T )) = T and ∀M =∑i∈I

δTi∈M (T) ,

M0(Paste (M)) =∑

i∈I :Ti =Υ

δTi.

(13)

The tree above a given level. Let (T, d, ρ) be a CLCR real tree and let a ∈ [0,∞). Recall thedefinition (Ti (a), d, σi (a)), i ∈ I (a), of the subtrees of T above a. We then set

Abv (a, T ) = ⊛i∈I (a)Ti (a). (14)

The tree Abv (a, T ) is called the tree above level a. By definition, the pointed isometry class ofAbv (a, T ) is ⊛i∈I (a)Ti (a) and it only depends on T . This induces a function from T to T that isdenoted in the same way and we check that

M0 Abv (a, ·) =Ma and Abv (a, ·) = Paste Ma . (15)

We also denote by Blw (a, T ) the tree (BT (ρ, a), d, ρ), that is the tree below the level a. Itspointed isometry class only depends on T : this induces a function from T to T that is denoted inthe same way.

Lemma 2.3. The map (a, T ) ∈ [0,∞) × T ↦−→ Abv (a, T ) is jointly continuous. The sameholds true for Blw .

Proof. See Appendix A.1.

Lemma 2.4. Fix a ∈ [0,∞) and h ∈ (0,∞). The following assertions hold true.

(i) Ma : T→M (T) is measurable and so are Z+a : T→ N ∪ ∞ and Z (h)a : T→ N.

(ii) Paste :M (T)→ T is measurable.


The functions D, ϑ and k. Let (T, d, ρ) be a CLCR real tree. Recall that ρ ∈ Br(T )∪Lf(T ), byconvention. We next set

D(T ) = inf d(ρ, σ ) ; σ ∈ Br(T ) ∪ Lf(T ) (16)

with the convention that inf ∅ = ∞. Namely, it is important to note that D(T ) = ∞ iff eitherT is reduced to a point or T is equivalent to a finite number of half lines pasted at their finiteendpoint. Recall from (9) the definition of Z+a and from (14) that of Abv (a, ·). If D(T ) < ∞,we set

ϑT := Abv (D(T ) , T ) and k(T ) = Z+0 (ϑT ) . (17)

If D(T ) = ∞, we set ϑT = ρ and k(T ) = 0. Observe that D(T ), k(T ) and the pointedisometry class of ϑT only depend on the pointed isometry class of T . So they can be viewed as



functions on T. With a slight abuse of language, D(T ) is viewed as the distance from the firstbranch point, ϑT as the tree above the first branch point and k(T ) + 1 as the degree of the firstbranch point.

Lemma 2.5. We have D = limh→0 D Rh . Moreover, the functions D : T → [0,∞],k : T→ N ∪ ∞ and ϑ : T→ T are measurable.


The functions D, ϑ and k are useful only when applied to discrete real trees and they allow tocharacterise T∗. More precisely, for every n ∈ N, we recursively define ϑn and Dn , ϑn+1 = ϑ ϑn

(where ϑ0 is taken as the identity map on T) and Dn = D ϑn . Thus, D = D0 and ϑ = ϑ1.

Lemma 2.6. We have that T∗ =T ∈ T :

∑n∈NDn(T ) = ∞

is a Borel subset of T.


2.2. Hereditary properties and their reduction procedures

Here, we generalise h-erasure Rh(T ) of a CLCR real tree (T, d, ρ) to reduction for moregeneral hereditary properties as explained in the introduction. More precisely, let (T, d, ρ) be aCLCR real tree and let σ ∈ T . Recall that the subtree above σ is given by

θσT =σ ′ ∈ T : σ ∈ [[ρ, σ ′]]

. (18)

Note that (θσT, d, σ ) is a CLCR real tree. We simply denote by θσT its pointed isometry class.Observe that

∀σ ′ ∈ [[ρ, σ ]] , θσT = θσ (θσ ′T ) . (19)

Definition 2.7.

(a) Let A ⊂ T be a Borel subset of T such that Υ ∈ A. The set A is hereditary if it satisfiesthe following: for every CLCR real tree (T, d, ρ) and for every σ ∈ T , if θσT ∈ A, thenT ∈ A.

(b) Let (T, d, ρ) be a CLCR real tree and let A ⊂ T be hereditary. We denote by RA(T ) theclosure in T of the subset

ρ ∪σ ∈ T : θσT ∈ A

. (20)

Then, (RA(T ), d, ρ) is a CLCR real tree that we call the A-reduced tree of T .

Let us briefly explain why (RA(T ), d, ρ) is a CLCR real tree. Since RA(T ) is a closed subset of T ,we only need to prove that it is connected. Denote by S the set given by (20). Let σ ∈ RA(T )\ρ(if any). Then, there exists a sequence σn ∈ S converging to σ and such that θσn T ∈ A. Letσ ′ ∈ [[ρ, σ [[ . For all sufficiently large n, σ ′ ∈ [[ρ, σn[[ , and (19) implies that θσ ′T ∈ A. Thisproves that [[ρ, σ ]] ⊆ RA(T ), for all σ ∈ RA(T ).

Clearly, the pointed isometry class of (RA(T ), d, ρ) only depends on T and A. This theninduces a function on T that is denoted in the same way by RA : T→ T. Note that RA(Υ ) = Υ .



Lemma 2.8. Let A ⊆ T be a hereditary property. Then RA : T→ T is Borel-measurable.

Proof. See Appendix B.2.

Lemma 2.9. Let (T, d, ρ) be a CLCR real tree and let A ⊆ T be hereditary. Then, the followingholds true:

∀σ ∈ RA(T ) , θσ RA(T ) = RA(θσT ). (21)

Proof. Set S = σ ′ ∈ θσT : θσ ′T ∈ A. Then, RA(θσT ) is the closure of σ ∪ S. Note thatif σ ∈ RA(T ), then RA(T ) ∩ θσT = θσ RA(T ). Thus, σ ∪ S ⊆ θσ RA(T ), which implies thatRA(θσT ) ⊆ θσ RA(T ). Conversely, let σ ′ ∈ RA(T )∩θσT be distinct from σ (if any). By definitionof RA(T ), there exists a sequence σn converging to σ ′ such that θσn T ∈ A. Since σ ∈ [[ρ, σ ′[[ ,then for all sufficiently large n we get σn ∈ θσT , and thus σn ∈ S. This proves that RA(T )∩ θσTis contained in the closure of σ ∪ S, which completes the proof of (21).

Let us now discuss how the tree reduction behaves with respect to the functions Ma , Abv (a, ·)and Z+a . To that end, for every hereditary property A ⊆ T we introduce

A− =T ∈ T : RA(T ) = Υ

, (22)

that can be viewed as a regular version of A.

Example 2.10. For all h ∈ [0,∞), consider the hereditary property Ah = Γ ≥ h. Notethat RAh is the h-leaf length erasure function as defined by (6). Namely RAh = Rh . Thus,A−h = Γ > h.

Lemma 2.11. Let A ⊆ T be hereditary. Then, the following holds true.

(i) A− ⊆ A and A− is hereditary. Moreover, RA− (T ) = RA(T ), for every CLCR real tree(T, d, ρ). This implies that (A−)− = A−.

(ii) Let a ∈ [0,∞) and let T ∈ T. If Ma(T ) =∑

i∈I δTi, then we get

Ma(RA(T )) =∑

i∈I :Ti∈A−

δRA(Ti ) and Z+a (RA(T )) = ⟨Ma(T ), 1A−⟩. (23)

Moreover,

⊛i∈I :

Ti∈A−

RA(Ti ) = Abv (a, RA(T )) = RA(Abv (a, T )). (24)

Proof. We first prove (i). By Lemma 2.8, A− is a Borel subset of T. We next prove that A− ⊆ A:indeed, let (T, d, ρ) be a CLCR real tree such that T ∈ A−; then, there is σ ∈ T \ ρ, such thatθσT ∈ A, which implies T ∈ A. Let us prove that A− is hereditary. Assume that θσT ∈ A−, thenthere exists σ ′ ∈ θσT \ σ such that θσ ′T ∈ A, which implies that RA(T ) = Υ , and T ∈ A−,which entails that A− is hereditary.

Let us prove that RA− (T ) = RA(T ). Since A− ⊆ A, RA− (T ) ⊆ RA(T ). Denote by S the setdefined in (20) and take σ ∈ S \ ρ (if any). Then, for all σ ′ ∈ [[ρ, σ [[ , σ ∈ RA(θσ ′T ), whichimplies that θσ ′T ∈ A− and therefore σ ′ ∈ RA− (T ). Thus, S is in the closure of RA− (T ), whichimplies the desired result.



We next prove (ii). Let (T, d, ρ) be a CLCR real tree. We denote by T i , i ∈ I , the connectedcomponents of the open set σ ∈ T : d(ρ, σ ) > a. For every i ∈ I , denote by σi the uniquepoint of T such that d(ρ, σi ) = a and such that Ti := σi ∪ T i is the closure of T i . Recall thatthe CLCR real trees (Ti , d, σi ) are the subtrees above a, namely: Ma(T ) =

∑i∈I δTi

. Let i ∈ Ibe such that Ti ∈ A−. Thus, RA(Ti ) does not reduce to σi and RA(Ti ) \ σi is connected andnon-empty. Namely, it is a connected component of σ ∈ RA(T ) : d(ρ, σ ) > a. Conversely,every connected component of σ ∈ RA(T ) : d(ρ, σ ) > a is of this form. This easily completesthe proof of (ii).

Recall that for leaf-length erasure Rh+h′ = Rh Rh′ . We extend these relations by introducingthe composition of hereditary properties.

Definition 2.12. Let A, A′ ⊆ T be two hereditary properties. The composition of A by A′ isdefined as the set A′ A := T ∈ T : RA(T ) ∈ A′.

Lemma 2.13. Let A, A′ ⊆ T be hereditary. Then, A′ A is hereditary and for every CLCR realtree (T, d, ρ), we get RA′ (RA(T )) = RA′A(T ).

Proof. Lemma 2.8 implies that A′ A is a Borel subset of T. Let (T, d, ρ) be a CLCR real treeand let σ ∈ T . By Lemma 2.9,

θσT ∈ A′ A ⇐⇒ σ ∈ RA(T ) and θσ RA(T ) ∈ A′. (25)

Note that if θσ RA(T ) ∈ A′, then RA(T ) ∈ A′, namely T ∈ A′ A. Then, “⇒” in (25) impliesthat A′ A is hereditary. Moreover, (25) immediately implies RA′ (RA(T )) = RA′A(T ).

2.3. Galton–Watson trees as random real trees

In this section, we give an intrinsic definition of Galton–Watson trees as T-valued randomvariables. Informally, given ξ = (ξ (k))k∈N, a probability measure on N, and c ∈ (0,∞), aGalton–Watson tree with offspring distribution ξ and lifetime parameter c is the genealogicaltree of a population that has a single progenitor and evolves as follows: the lifetimes ofthe individuals are independent and exponentially distributed with mean 1/c ; when they die,individuals independently give birth to a random number of children distributed according to ξ .We denote by ϕξ the generating function of ξ and by mξ the mean of ξ . Namely,

∀ r ∈ [0, 1] :∑k≥0

r kξ (k) = ϕξ (r ) and mξ =

∑k≥0

kξ (k) = ϕ′ξ (1−) ∈ [0,∞]. (26)

We make the following assumptions on the offspring distribution ξ :non-tr ivial : ξ (0)+ ξ (1) < 1, proper : ξ (1) = 0,

conservative :∫ 1− dr

(ϕξ (r )− r )−= ∞ ,

(27)

where ( · )− stands for the negative part. We assume that ξ is proper because we are onlyinterested in the underlying geometrical tree. The conservativity assumption comes from astandard criterion that asserts that the right profile of a Galton–Watson tree is an N-Markovprocess that is conservative iff the last condition of (27) is satisfied (see [5, Section III.2,Theorem 1] for more details).



Recall from Section 2 that for all T1, T2 ∈ T, T1 ⊛ T2 stands for the pointed isometry classof the tree obtained by pasting two representatives of T1 and T2 at their roots. We let the readercheck that (T1, T2) ↦→ T1 ⊛ T2 is a continuous, symmetric and associative operation on T. Forall Borel probability measures Q1 and Q2 on T, we define Q1 ⊛ Q2 as the image measure of theproduct measure Q1 ⊗ Q2 under ⊛. Namely, for all measurable G : T→ [0,∞),∫

T(Q1 ⊛ Q2)(dT ) G(T ) =

∫T

∫T

Q1(dT1)Q2(dT2) G(T1 ⊛ T2) .

We easily check that (Q1, Q2) ↦→ Q1 ⊛ Q2 is a weakly continuous, symmetric and associativeoperation. For all Borel probability measures Q on T, and all n ∈ N, we recursively define Q⊛n

by setting Q⊛(n+1)= Q ⊛ Q⊛n where Q⊛0 is the Dirac mass at the point tree Υ . Recall from

Section 2 the definition of D, ϑ and k.

Definition 2.14. Let ξ be a proper and conservative offspring distribution. Let c ∈ (0,∞).

(a) A Galton–Watson real tree with offspring distribution ξ and edge parameter c (a GW(ξ, c)-real tree for short) is a T-valued random variable whose distribution Q satisfies thefollowing. For all n ∈ N, for all measurable functions g : [0,∞) → [0,∞) andG : T→ [0,∞),

Q[1k=nG(ϑ)g(D)

]= ξ (n)Q⊛n[ G ]

∫∞

0g(t)ce−ct dt. (28)

(b) A Galton–Watson real forest with offspring distribution ξ , edge parameter c and initialdistribution µ (a GW(ξ, c;µ)-real forest for short) is a T-valued random variable whosedistribution is P =

∑n≥0µ(n)Q⊛n , where Q stands for the distribution of a GW(ξ, c)-real

tree.

Lemma 2.15. For each proper conservative offspring distribution ξ and each c ∈ (0,∞),there exists a unique distribution Qξ,c on T that satisfies (28) in Definition 2.14(a). Moreover,Qξ,c(T∗) = 1 and (ξ, c) ↦→ Qξ,c is injective.

Proof. See Appendix C.1.

Notation. For all proper and conservative distributions ξ , for all c ∈ (0,∞) and for all probabilitymeasures µ on N, we set Pξ,c,µ =

∑n≥0µ(n)Q⊛n

ξ,c as the law of a GW(ξ, c;µ)-real forest.

In the non-conservative case Qξ,c does not exist as a distribution on T. In fact, explosive GW-real trees would not be locally compact. If we were to consider them up to the first explosiontime, the tree would be locally compact, but property (28) would fail, because the explosiontimes of the children of the ancestor are distinct.

GW-real trees are characterised by their regenerative branching property as proved byWeill [42] in the compact case. More precisely, for every a ∈ [0,∞), we denote by Ba thesigma-field on T generated by Blw (a, ·). Note that (Ba)a∈[0,∞) is a filtration on T. We denote by(Ba+)a∈[0,∞) the associated right-continuous filtration. Recall from (9) the definition of the rightprofile at height a that is denoted by Z+a . We easily check that Z+a is Ba+-measurable.

Lemma 2.16. Let Q be a Borel probability measure on T. We first assume that Q(0 < D <

∞) > 0. We also assume that Q-a.s. the process a ↦→ Z+a is N-valued and cadlag . Then, thefollowing assertions are equivalent.



(i) For every a ∈ [0,∞), the conditional distribution given Z+a of Abv (a, ·) under Q is Q⊛Z+a .(ii) For every a ∈ [0,∞), the conditional distribution given Ba+ of Abv (a, ·) under Q is

Q⊛Z+a .(iii) Q is the distribution of a GW(ξ, c)-real tree, for a certain c ∈ (0,∞) and a certain proper

conservative offspring distribution ξ .

Proof. In the compact case, this statement is close to [42, Theorem 1.2]. We briefly proveLemma 2.16 in Appendix C.2 using different arguments. We also mention that the cadlagassumption can be dropped, as we show in Theorem 3.21.

Let us recall basic results on the branching processes associated with GW-trees. Let c ∈(0,∞) and let ξ be a non-trivial proper conservative offspring distribution. The regenerativebranching property entails that (Z+a )a∈[0,∞) under Qξ,c is an N-valued Markov process whosematrix-generator (qi, j )i, j∈N is given by qi, j = c i ξ ( j − i + 1), if j > i or j = i − 1, qi,i = −c iand qi, j = 0 if j < i − 1. Let us set

w(a, θ) := Qξ,c

[e−θ Z+a

], a, θ ∈ [0,∞). (29)

The definition of GW-real trees implies that for all θ ∈ [0,∞), a ↦→ w(a, θ) is the uniquenonnegative solution of the differential equation ∂w(a, θ)/∂a = c

(ϕξ (w(a, θ))− w(a, θ)

), with

w(0, θ) = e−θ . A simple change of variables allows to rewrite the previous equation in thefollowing form:

∀θ ∈ (0,∞) \ − log qξ , ∀a ∈ [0,∞) ,∫ w(a,θ )

e−θ

drϕξ (r )− r

= ca , (30)

where qξ is the smallest solution of the equation ϕξ (r ) = r and where we agree on − log(qξ ) =∞ if qξ = 0 (see e.g. [5, Section III.3] for more details). Note that Qξ,c-a.s. Γ = infa ∈[0,∞) : Z+a = 0. Thus,

w(a) := Qξ,c(Γ ≤ a) = limθ→∞↓ w(a, θ) satisfies

∫ w(a)

0

drϕξ (r )− r

= ca. (31)

Let µ be a probability measure on N that is distinct from δ0. We easily derive from Lemma 2.16that for all a ∈ [0,∞),

the conditional distribution given Z+a of Abv (a, ·) under Pξ,c,µ is Q⊛Z+aξ,c . (32)

Then, under Pξ,c,µ, (Z+a )a∈[0,∞) is also a Markovian branching process with offspring distributionξ , with lifetime parameter c and with initial distribution µ. Let ϕµ be the generating function ofµ. We thus get

∀a, θ ∈ [0,∞) , Pξ,c,µ[e−θ Z+a

]= ϕµ (w(a, θ)) . (33)

Lemma 2.17. Let (ξn)n≥0, be a sequence of proper conservative offspring distributions thatconverges weakly to the proper conservative offspring distribution ξ∞. Let (cn)n≥0 be a sequenceof positive real numbers that converges to c∞ ∈ (0,∞). Let (µn)n≥0 be a sequence of probabilitymeasures on N that converges weakly to the probability measure µ∞ on N. Then, the followingconvergences hold weakly on T:

limn→∞

Qξn ,cn = Qξ∞,c∞ and limn→∞

Pξn ,cn ,µn = Pξ∞,c∞,µ∞ .



Moreover, for all a ∈ [0,∞), as n → ∞, the law of Z+a under Qξn ,cn (resp. under Pξn ,cn ,µn )converges weakly to the law of Z+a under Qξ∞,c∞ (resp. under Pξ∞,c∞,µ∞ ).

Proof. See Appendix C.3.

The following theorem, which is the main result of this section, shows that the class of GW-laws is stable under reduction by a hereditary property. Recall that Qξ,c stands for the law ofa GW(ξ, c)-real tree, that ϕξ is the generating function of ξ and that qξ is the smallest root ofϕξ (r ) = r , which is the extinction probability of a GW(ξ, c)-real tree.

Theorem 2.18. Let ξ be a non-trivial proper conservative offspring distribution and letc ∈ (0,∞). Let A ⊆ T be hereditary. We assume that

α := Qξ,c(RA = Υ ) ∈ (0, 1). (34)

Then, ξ (0) > 0, α ∈ (0, qξ ] and thus ϕ′ξ (α) ∈ (0, 1). Moreover, RA under the conditioned lawQξ,c( · | RA = Υ ) is distributed as a GW(ξ (α), c(α))-real tree where

c(α)= c (1− ϕ′ξ (α)) and ϕξ (α) (r ) = r +

ϕξ (α + (1− α)r )− α − (1− α)r(1− α)(1− ϕ′ξ (α))

,

r ∈ [0, 1]. (35)

Proof. First recall from (22) that RA = Υ = A− and thus α = Qξ,c(T \ A−). Recall from(16) and (17) the definition of the functions D, ϑ and k. Since A− is hereditary, we get 1T\A− ≤1k=0 + 1k≥11⟨MD (·),1A− ⟩=0. We take the expectation under Qξ,c: since MD =M0 ϑ , (28)in the definition of Qξ,c entails that

0 < α ≤ ξ (0)+∑n≥1

ξ (n)Q⊛nξ,c ( ⟨M0, 1A−⟩ = 0 ) =

∑n≥0

ξ (n)αn= ϕξ (α) ,

which entails that α ∈ (0, qξ ] and that ϕ′ξ (α) ∈ (0, 1).We denote by Q the law of RA under Qξ,c( · | A−) and we want to apply Lemma 2.16 to Q.

To that end, we first note that a ↦→ Z+a is N-valued and cadlag Q-a.s. and then claim that also

Q(0 < D <∞) = Qξ,c(

0 < D RA <∞| A−)> 0. (36)

Indeed, let T ∈ T∗ be such that Z+0 (T ) = 1 and D(T ) ∈ (0,∞). If we have ⟨MD(T )(T ), 1A−⟩ ≥

2, then D(RA(T )) = D(T ) and since A− is hereditary, we also get T ∈ A−. Consequently,

x := Qξ,c (⟨MD, 1A−⟩ ≥ 2) ≤ Qξ,c(

A− ∩ D RA ∈ (0,∞))

= (1− α)Q(0 < D <∞). (37)

Now by (28) in the definition of Qξ,c, we get

x =∑n≥2

ξ (n)Q⊛nξ,c (⟨M0, 1A−⟩ ≥ 2)

=

∑n≥2

ξ (n)(1− Q⊛n

ξ,c ( ⟨M0, 1A−⟩ = 0 )− Q⊛nξ,c ( ⟨M0, 1A−⟩ = 1 )

)=

∑n≥2

ξ (n)(1− αn

− n(1− α)αn−1)= 1− ϕξ (α)− (1− α)ϕ′ξ (α) > 0 ,

since ϕξ is strictly convex (because ξ is proper and non-trivial). Then, (37) implies (36).



Recall that under Qξ,c and conditionally given Z+a , Abv (a, ·) has law Q⊛Z+aξ,c . Recall (23) and

(24) and observe that for all T ∈ T∗ and for all a ∈ [0,∞), Z+a (RA(T )) ≥ 1, implies T ∈ A−.Thus, for all n ≥ 1, and for all measurable functions G : T→ [0,∞), we get

Bn := Q(

1Z+a =nG(Abv (a, ·))

)=

11− α

Qξ,c

(1Z+a RA=nG(Abv (a, RA(·)))

)=

11− α

Qξ,c(1⟨M0(Abv (a,·)),1A− ⟩=nG(RA(Abv (a, ·)))

)=

11− α

∑N≥n

Qξ,c(Z+a = N )∫∑

1≤i≤N 1A− (Ti )=nQ⊗Nξ,c (dT1 . . . dTN )

×G

(⊛

1≤i≤N :Ti∈A−RA(Ti )

)

=1

1− α

∑N≥n

Qξ,c(Z+a = N )(

Nn

)αN−n(1− α)n

∫Tn

Q⊗n(dT ′1 . . . dT ′n)

×G(T ′1 ⊛ . . .⊛ T ′n

)=

11− α

∑N≥n

Qξ,c(Z+a = N )(

Nn

)αN−n(1− α)n Q⊛n [G] . (38)

This proves that under Q, the law of Abv (a, ·) conditionally given Z+a is Q⊛Z+a . By Lemma 2.16,Q is the law of a GW(ξ∗, c∗)-real tree where ξ∗ is a proper conservative offspring distribution.We next set w∗(a, θ) = Q[exp(−θ Z+a )] and w(a, θ) = Qξ,c[exp(−θ Z+a )]. By summing (38)over n ≥ 1 with G = 1− e−θ Z+0 , we easily get

w∗(a, θ) = 1−1− w

(a ,− log(α + (1− α)e−θ )

)1− α

. (39)

and by differentiating (30) with respect to a and to θ , we get

−∂aw∗(a, θ)∂θw∗(a, θ)

= c∗eθ(ϕξ∗ (e

−θ )− e−θ).

On the other hand, (39) implies that

−∂aw∗(a, θ)∂θw∗(a, θ)

= −α + (1− α)e−θ

(1− α)e−θ·∂aw(a,− log(α + (1− α)e−θ ))∂θw(a,− log(α + (1− α)e−θ ))

=ceθ

1− α

(ϕξ (α + (1− α)e−θ )− α − (1− α)e−θ

).

Thus, we get

c∗(ϕξ∗ (e

−θ )− e−θ)=

c1− α

(ϕξ (α + (1− α)e−θ )− α − (1− α)e−θ

). (40)

Since ξ∗ is proper, ξ∗(1) = ϕ′ξ∗ (0) = 0. Thus, by differentiating (40) and by letting θ go to∞,we find c∗ = c(α), as defined in (35). This, combined (40), implies ξ∗ = ξ (α).

The previous result and (24) in Lemma 2.11 applied to a = 0, immediately imply thefollowing statement for forests. Recall that Pξ,c,µ =

∑n∈Nµ(n)Q⊛n

ξ,c .

Theorem 2.19. Let ξ be a proper non-trivial conservative offspring distribution, let c ∈ (0,∞)and let µ be a probability distribution on N such that µ(0) < 1. Let A ⊆ T be hereditary. We set



α = Qξ,c(RA = Υ ) and we assume that α ∈ (0, 1). Then, RA under Pξ,c,µ has law Pξ (α),c(α),µα ,where ξ (α) and c(α) are given by (35) and µα is given by

ϕµα (r ) = ϕµ(α + (1− α)r ) , r ∈ [0, 1]. (41)

3. Growth processes

3.1. Preliminaries on convergence criteria

This section is devoted to general convergence theorems for complete locally compact rootedreal trees (CLCR real trees): we first state a general tightness criterion involving the h-erasedprofiles and we also state an almost sure convergence criterion for sequences of trees that “grow”in a broad sense that we will make precise later.

We use the following notation. Let (T, d, ρ) be a rooted real tree, not necessarily locallycompact. Then, for every h ∈ (0,∞) the definition (6) of h-leaf-erased tree (Rh(T ), d, ρ) makessense and it defines a path-connected subset of T . It is therefore a rooted real tree. For everya ∈ [0,∞), we define Z (h)

a (T ) as in (8). Note that this quantity may be infinite if T is not locallycompact. We denote by NT (h, r ) the (possibly infinite) minimal number of open balls with radiush that are necessary to cover BT (ρ, r ).

Lemma 3.1. Let (T, d, ρ) be a complete rooted real tree (not necessarily locally compact). Then,

∀a, h ∈ (0,∞) , ∀r ∈ (a + h,∞) , Z (h)a (T ) ≤ NT (h, r ). (42)

Moreover, let D ⊆ (0,∞) be a dense subset. Then, for every h, r ∈ (0,∞), there is a finitesubset Dh,r ⊂ D ∩ [0, r ] such that

NT (h, r ) ≤ 1+∑

a∈Dh,r

Z (h/3)a (T ). (43)

Consequently, (T, d, ρ) is locally compact iff there exist a dense subset D ⊆ (0,∞) and asequence (h p)p∈N decreasing to 0 such that

∀a ∈ D , ∀p ∈ N , Z (h p)a (T ) <∞. (44)

Proof. Let a, h ∈ (0,∞) and r > a + h. We fix L = σ ∈ Rh(T ) : d(ρ, σ ) = a so thatZ (h)

a (T ) = #L . Note that NT (h, r ) ≥ 1. Thus, (42) holds for #L ≤ 1. Now assume that #L ≥ 2.With every σ ∈ L , we can associate ζ (σ ) ∈ T such that d(ρ, ζ (σ )) = a + h and σ ∈ [[ρ, ζ (σ )]].Note that ζ (σ ) ∈ BT (ρ, r ). Suppose that σ and σ ′ are two distinct points of L . Recall the notationσ ∧ σ ′ for the most recent common ancestor of σ and σ ′. Then, d(ζ (σ ), ζ (σ ′)) > 2h becauseζ (σ ) ∧ ζ (σ ′) = σ ∧ σ ′ is below level a in T . Thus, Z (h)

a (T ) = #L ≤ NT (h, r ).Let h ∈ (0,∞) and let D ⊂ (0,∞) be dense. We can find an increasing sequence a(k) ∈ D,

k ∈ N, such that a(0) < h/2 and h ≤ a(k + 1) − a(k) < 3h/2. For every k ∈ N, we thenset Lk = σ ∈ Rh(T ) : d(ρ, σ ) = a(k). Let σ ∈ T be such that d(ρ, σ ) ≥ 2h. Thereexists k ∈ N such that a(k + 1) ≤ d(ρ, σ ) < a(k + 2). Thus, [[ρ, σ ]] ∩ Lk = σ

′ and

d(σ, σ ′) < a(k + 2) − a(k) < 3h. This implies NT (3h, r ) ≤ 1 +∑

0≤k≤⌊r/h⌋Z(h)a(k)(T ), where

⌊r/h⌋ stands for the integer part of r/h. This easily entails (43).If T is locally compact, we already noticed that Z (h)

a (T ) < ∞, for every a, h ∈ (0,∞).Conversely, suppose that (44) holds true. Note that h ↦→ Z (h)

a (T ) is non-increasing. Then, (43)applies and for every r , and (BT (ρ, r ), d) is a uniformly bounded complete metric space. It istherefore compact. This implies that T is locally compact.



Note that NT (h, r ) only depends on the isometry class T of T , which justifies the notationNT (h, r ). Let C ⊆ T. As a consequence of a general result on compactness with respect to thepointed Gromov–Hausdorff metric δcpct (see [22] or [7, Theorem 8.1.10]), we have the followingresult.

The δ-closure of C is compact ⇐⇒ ∀h, r ∈ (0,∞) , supT∈C

NT (h, r ) <∞. (45)

Thus, Lemma 3.1 immediately entails the following theorem.

Theorem 3.2. Let C ⊆ T. Then, the δ-closure of C is compact iff there exist a sequence (h p)p∈Ndecreasing to 0 and a countable dense subset D ⊂ (0,∞) such that

∀p ∈ N , ∀a ∈ D , supT∈C

Z (h p)a (T ) <∞ . (46)

The same statement holds true when Z (h p)a is replaced by Z (h p)

a+ .

Similar ideas have been used to obtain criteria in terms of related quantities such as totallengths of trees (see [16, Lemma 2.6]). The following tightness criterion for random trees is aconsequence of Theorem 3.2.

Theorem 3.3. Let (T j ) j∈J be family of T-valued random trees. Their laws are δ-tight if andonly if for every fixed h, a ∈ (0,∞), the laws of the N-valued random variables (Z (h)

a (T j )) j∈J

are tight. The same result holds true when (Z (h)a (T j )) j∈J is replaced by (Z (h)

a+(T j )) j∈J .

Proof. Suppose that for every fixed h, a ∈ (0,∞) the laws of (Z (h)a (T j )) j∈J are tight. Let

D = aq; q ∈ N be some dense subset of (0,∞) and let (h p)p∈N be a sequence decreasingto 0. Let ε ∈ (0, 1). For all p, q ∈ N, there exists K p,q ∈ (0,∞) such that

supj∈J

P(Z (h p)aq (T j ) > K p,q ) < ε2−p−q−2 .

We then set C = T ∈ T : ∀p, q ∈ N, Z (h p)aq (T ) ≤ K p,q. Theorem 3.2 implies that its δ-closure

C is compact and we easily prove that for every j ∈ J

P(T j ∈ C) ≤ P(∃p, q ∈ N : Z (h p)aq (T j ) > K p,q ) ≤

∑p,q∈N

P(Z (h p)aq (T j ) > K p,q ) < ε ,

which entails the tightness of the laws of T j , j ∈ J . Conversely, let ε ∈ (0, 1). There exists aδ-compact subset K ⊂ T such that inf j∈J P(T j ∈ K) ≥ 1− ε. Fix a, h ∈ (0,∞), r > a + h, andset K = supT∈KNT (h, r ). By (45), K < ∞ and Lemma 3.1 implies that supT∈KZ (h)

a (T ) ≤ K .Thus, inf j∈J P(Z (h)

a (T j ) ≤ K ) ≥ 1−ε, which implies the tightness of the laws of Z (h)a (T j ), j ∈ J ,

for each a, h ∈ (0,∞).

Corollary 3.4. Let (Tn)n∈N be a sequence of T-valued random variables. We suppose that forevery sufficiently small h ∈ (0,∞), the laws of Rh(Tn), n ∈ N, converge weakly in (T, δ) asn→∞. Then, the laws of the Tn , n ∈ N, converge weakly in (T, δ) as n→∞.

Proof. Fix a, h ∈ (0,∞). Since the laws of Rh/2(Tn), n ∈ N, converge weakly, Theorem 3.3implies that the laws of Z (h/2)

a (Rh/2(Tn)) = Z (h)a (Tn), n ∈ N, are tight. Theorem 3.3 then entails

that the laws of Tn , n ∈ N, are δ-tight. Denote by Λ1 and Λ2 two limit laws of (Tn)n∈N. Then,



the laws of Rh under Λ1 and Λ2 coincide, since Rh is δ-continuous and since we assume that thelaws of Rh(Tn), n ∈ N, converge weakly. Thus, Λ1 = Λ2 since Rh converges uniformly to theidentity on T, as h decreases to 0.

We now discuss a stronger convergence for sequences of CLCR real trees that grow in thefollowing weak sense.

Definition 3.5. Let T , T ′ ∈ T. We say that T can be embedded in T ′, which is denoted byT ⪯ T ′ if we can find representatives (T, d, ρ) and (T ′, d ′, ρ ′) of T and T ′, and an isometricalembedding f : T → T ′ such that f (ρ) = ρ ′.

With this definition, we have for any hereditary property A ⊆ T that

∀T ∈ T , RA(T ) ⪯ T . (47)

We easily check that ⪯ is a partial order on T (anti-symmetry property is the only non-trivialpoint to check). Note that if T ⪯ T ′, then for all a ∈ [0,∞) and for all h ∈ (0,∞),

Z+a (T ) ≤ Z+a (T ′) , Rh(T ) ⪯ Rh(T ′) and Z (h)a (T ) ≤ Z (h)

a (T ′). (48)

We first prove the following convergence criterion for a growing sequence of trees.

Theorem 3.6. Let (Tn)n∈N be a T-valued sequence such that Tn ⪯ Tn+1, n ∈ N. Let D ⊆ (0,∞)be dense and let (h p)p∈N be a sequence decreasing to 0. We assume the following:

∀p ∈ N , ∀a ∈ D , supn≥0

Z (h p)a (Tn) < ∞. (49)

Then, there exists T ∈ T such that limn→∞δ(Tn, T ) = 0.

The proof of the theorem is in several steps. Representation lemmas are well-known (seee.g. [19, Lemma A.1]). The following version furthermore represents ⪯-increasing sequences asinclusions in a common metric space:

Lemma 3.7. Let (Tn)n∈N be a T-valued sequence such that Tn ⪯ Tn+1, n ∈ N. Then, there exista pointed Polish space (E, d, ρ) and a sequence Tn ⊆ E such that for all n ∈ N

ρ ∈ Tn , Tn ⊆ Tn+1 and (Tn, d, ρ) is a representative of Tn .

Proof. We first prove recursively that for all n ∈ N, there is a representative (T ∗n , dn, ρn) of Tn

and an isometrical embedding fn : T ∗n → T ∗n+1 such that fn(ρn) = ρn+1. Indeed, assume that(T ∗0 , d0, ρ0), . . . , (T ∗n , dn, ρn) and f0, . . . , fn−1 exist and satisfy the above mentioned conditions.Since Tn ⪯ Tn+1, there are (T ′n, d ′n, ρ

′n) and (T ∗n+1, dn+1, ρn+1) that are representatives of Tn

and Tn+1, respectively, and there is an isometrical embedding f ′n : T ′n → T ∗n+1 such thatf ′n(ρ ′n) = ρn+1. There also exists an isometry φ : T ∗n → T ′n such that φ(ρn) = ρ ′n . Then, weset fn = f ′n φ that satisfies the desired property, which completes the recurrence.

We next construct the desired spaces in a projective way. To that end, we first set

∀n ∈ N , Sn :=

σ = (σk)k∈N ∈

∏k∈N

T ∗n+k : σk+1 = fn+k(σk)

and S∞ =

⨆n∈N

Sn ,



where ⊔ stands for the disjoint union. For all σ ∈ T ∗n , we also define the sequence ȷn(σ ) =(σk)k∈N such that σ0 = σ and σk+1 = fn+k(σk), k ∈ N. Note that ȷn is one-to-one from T ∗n ontoSn . Next, observe that for all σ = (σk)k∈N ∈ Sn and all σ ′ = (σ ′k)k∈N ∈ Sn′ ,

∀k ≥ |n′ − n| , dn+k(σk, σ′

n−n′+k) = dn′+k(σn′−n+k, σ′

k) =: d(σ , σ ′) .

This induces a pseudo-metric d on S∞ and we see that d(σ , σ ′) = 0 iff a certain shift of thesequence σ is equal to a certain shift of the sequence σ ′. We now introduce the usual equivalencerelation ≡ by specifying that σ ≡ σ ′ iff d(σ , σ ′) = 0. We denote by E∞ = S∞/ ≡ the quotientspace and we denote by π : S∞ → E∞ the canonical projection that associates with a point σ

its equivalence class. Then, d induces a true metric on E∞ that is denoted in the same way (tosimplify notation). We next set:

ρ = π (ȷ0(ρ0)) and Tn = π (Sn) , n ∈ N .

First note that ȷn(ρn) ≡ ȷ0(ρ0). Thus, ρ ∈ Tn . We next check that Tn ⊆ Tn+1. Indeed, letσ ∈ T ∗n and set ȷn(σ ) = (σk)k∈N. Then, σ1 ∈ T ∗n+1 and ȷn+1(σ1) = (σk+1)k∈N ∈ Sn+1; thusȷn(σ ) ≡ ȷn+1(σ1) and π (ȷn(σ )) ∈ Tn+1, which entails Tn ⊆ Tn+1.

For all n ∈ N, we then set φn = π ȷn : T ∗n → E∞. It is easy to check that φn(ρn) = ρ andthat φn is an isometry from T ∗n onto Tn . Therefore (Tn, d, ρ) is a CLCR real tree whose pointedisometry class is Tn . Observe that E∞ =

⋃n∈NTn , which proves that (E∞, d∞) is separable. We

get the desired result by taking E as a completion of (E∞, d) such that E∞ ⊆ E .

Let (E, d, ρ) be a pointed Polish space. For the following result, we introduce the set of CLCRreal trees embedded in (E, d, ρ).

TE = T ⊆ E : ρ ∈ T and (T, d, ρ) is a CLCR real tree. . (50)

Let us denote by dHaus the Hausdorff distance on compact subsets of E . Then, for any T, T ′ ∈ TE ,we define

dE (T, T ′) =∑k≥1

2−kdHaus(BT (ρ, k), BT ′ (ρ, k)

).

Clearly, dE is a distance on TE and we have

δ(T, T ′) ≤ dE (T, T ′). (51)

We next prove the following criterion for convergence in TE .

Lemma 3.8. Let (E, d, ρ) be a pointed Polish space. Let (Tn)n∈N be a TE -valued sequence suchthat Tn ⊆ Tn+1. Let D ⊆ (0,∞) be dense and let (h p)p∈N be a sequence decreasing to 0. Weassume the following.

∀p ∈ N , ∀a ∈ D , supn≥0

Z (h p)a (Tn) <∞. (52)

Let T be the closure of⋃

n∈NTn in (E, d). Then,

T ∈ TE and limn→∞

dE (Tn, T ) = 0 .

Proof. We first set T∞ =⋃

n∈NTn , so that T is the closure of T∞ in E . The restriction of d toT∞ satisfies the four points condition. We check that the same holds true for the restriction of dto T . Moreover, every point of T∞ belongs to a certain Tn and is connected to ρ by a geodesic.This easily implies that (T, d) is a connected space. Thus, (T, d, ρ) is a Polish rooted real tree.



We fix h, a ∈ (0,∞) and we set La = σ ∈ R2h(T ) : d(ρ, σ ) = a. Thus, #La = Z (2h)a (T ).

Let σ ∈ La . There exists s ∈ T such that σ ∈ [[ρ, s]] and d(σ, s) ≥ 2h. Since T is the closureof T∞, there exist n(σ ) ∈ N and γ ∈ Tn(σ ) such that d(s, γ ) < h. This implies σ ∈ [[ρ, γ ]] andd(σ, γ ) > h. Namely, this implies that σ ∈ Rh(Tn(σ )) and thus σ ∈ Rh(Tn), for all n ≥ n(σ ). Wethen get

Z (2h)a (T ) = sup

n≥0#σ ∈ La : n(σ ) ≤ n ≤ sup

n≥0Z (h)

a (Tn) .

Note that this inequality holds true for all h, a ∈ (0,∞). By (52), Lemma 3.1 applies and(T, d, ρ) is a CLCR real tree, which shows that T ∈ TE .

Let us fix r, ε ∈ (0,∞). Since BT (ρ, r ) is compact and since T∞ is dense in T , we easilysee that there exist nε ∈ N and σ1, . . . , σp ∈ BTnε (ρ, r ) such that for all σ ∈ BT (ρ, r ),d(σ1, . . . , σp, σ ) < ε. For all n ≥ nε, since BTnε (ρ, r ) ⊆ BTn (ρ, r ) ⊆ BT (ρ, r ), we getdHaus(BTn (ρ, r ), BT (ρ, r )) < ε, which easily completes the proof of the lemma.

Proof of Theorem 3.6. Let (Tn)n∈N be a T-valued sequence that satisfies Tn ⪯ Tn+1 and theassumption (49). By Lemma 3.7, there exist a pointed Polish space (E, d, ρ) and Tn ∈ TE ,n ∈ N, that satisfy Tn ⊆ Tn+1, (52) and such that (Tn, d, ρ) is a representative of Tn . Then,Lemma 3.8 applies and there exists T ∈ TE such that limn→∞dE (Tn, T ) = 0, which implieslimn→∞δ(Tn, T ) = 0, by (51).

Theorem 3.6 entails the following criterion for growing sequences of random trees.

Theorem 3.9. Let (Tn)n∈N be a T-valued sequence of random trees such that for all n ∈ N,

P-a.s. Tn ⪯ Tn+1 .

We assume that for every fixed a, h ∈ (0,∞), the family of laws of N-valued random variablesZ (h)

a (Tn), n ∈ N, is tight. Then, there exists a random tree T in T such that

P-a.s. limn→∞

δ(Tn, T ) = 0 .

Proof. We fix a, h ∈ (0,∞). By (48), a.s. Z (h)a (Tn) ≤ Z (h)

a (Tn+1), for all n ∈ N. Then for allK ∈ (0,∞),

P(

supn∈N

Z (h)a (Tn) > K

)= sup

n∈NP(Z (h)

a (Tn) > K).

Since the family of laws of Z (h)a (Tn), n ∈ N, is tight, we get P(supn∈NZ (h)

a (Tn) < ∞) = 1, foreach a, h ∈ (0,∞), and Theorem 3.6 easily completes the proof.

Lemma 3.8 also entails the following almost sure convergence criterion for random trees inTE . The proof, quite similar to that of Theorem 3.9, is left to the reader.

Theorem 3.10. Let (E, d, ρ) be a pointed Polish space. Let (Tn)n∈N be a TE -valued sequenceof random trees such that a.s. Tn ⊆ Tn+1, n ∈ N. We assume that for each fixed a, h ∈ (0,∞),the family of laws of N-valued random variables Z (h)

a (Tn), n ∈ N, is tight. Let T be the closureof⋃

n∈NTn in E. Then,

P-a.s. T ∈ TE and limn→∞

dE (Tn, T ) = 0 .



3.2. Definition and characterisation of growth processes

3.2.1. Infinitely extensible offspring distributionsWe first briefly study the transform on offspring distributions that appears in (35) in

Theorem 2.18. Let ξ be a proper offspring distribution, let c ∈ (0,∞) and let µ be anotherprobability on N. Recall that ϕξ stands for its generating function and that qξ is the smallest rootof ϕξ (r ) = r . We introduce the following subset

Dξ = α ∈ [0, 1) : α ≤ ϕξ (α) , (53)

i.e. Dξ = [0, 1) if qξ = 1, and Dξ = [0, qξ ] if qξ < 1. We easily check that for all α ∈ Dξ ,ϕ′ξ (α) < 1, and we define (ξ (α), c(α), µα) by setting for all r ∈ [0, 1],

c(α)= c(1− ϕ′ξ (α)), ϕξ (α) (r ) = r +

ϕξ (α + (1− α)r )− α − (1− α)r(1− α)(1− ϕ′ξ (α))

,

ϕµα (r ) = ϕµ(α + (1− α)r ).(54)

Note that ξ (α) is proper. Then (ξ, c, µ) can be recovered from (ξ (α), c(α), µα) and α as follows.First note that ϕξ (α) can be analytically extended to ( −α1−α , 1). Observe that the right limits ofϕξ (α) and ϕ′

ξ (α) at −α1−α exist and are finite. Since ξ is proper we easily get (1 − ϕ′ξ (α))−1=

1− ϕ′ξ (α) (

−α1−α ) > 1. We therefore get for all r ∈ [0 , 1],

c = c(α)(1− ϕ′ξ (α) (−α

1− α)), ϕξ (r ) = r +

(1− α)ϕξ (α) ( r−α1−α )− (r − α)

1− ϕ′ξ (α) (

−α1−α )

,

ϕµ(r ) = ϕµα (r − α1− α

).

(55)

Next note that if β ∈ Dξ (α) , then((ξ (α))(β), (c(α))(β), (µα)β

)= (ξ (γ ), c(γ ), µγ )

with γ ∈ Dξ and 1− γ = (1− α)(1− β). (56)

We now introduce the definition of infinitely extensible offspring distributions.

Definition 3.11. An offspring distribution ξ is said to be infinitely extensible if there exists asequence of proper offspring distributions (ξn)n∈N and αn ∈ Dξn , n ∈ N, such that ξ = ξ (αn )

n forall n ∈ N, and limn→∞αn = 1.

The term infinitely extensible, which is new, has been coined by analogy with infinitelydivisible laws.

Let us make several remarks about this definition. If ξ (0) > 0 and if ξ = ξ (α), for all α ∈ Dξ ,we can show that ξ is a stable offspring distribution, namely, there exists γ ∈ (1, 2] such thatϕξ (r ) = r + 1

γ(1 − r )γ . Specifically, consider g(s) = ϕξ (1 − s) − (1 − s), then it is easy to

see from (35) that βg′(β)g(s) = g(sβ) for all s ∈ [0, 1] and β ∈ (1 − qξ , 1]; for s = 1, wecan solve the resulting differential equation to see g(β) = ξ (0)β1/ξ (0), hence with γ = 1/ξ (0),ϕξ (r ) = r+ 1

γ(1−r )γ for some γ > 1. Finally, this is only the generating function of an offspring

distribution if γ ∈ (1, 2]. Observe that stable offspring distributions are critical (i.e. their meanis equal to 1), but their variance is infinite except in the binary case γ = 2. Stable offspringdistributions occur in many contexts: see for instance Le Jan [34] and Neveu [36] for leaf-lengtherasure. See also [11], Vatutin [41] and Jakymiv [26] for reduced trees. We remark that there are



offspring distributions ξ that are not stable but such that ξ = ξ (α) holds true for certain α ∈ (0, 1).Note that any infinitely extensible offspring distribution is proper.

More generally, infinitely extensible offspring distributions are characterised by a branchingmechanism, that is, by a function ψ : [0,∞) → R that is of the following Levy–Khintchineform

ψ(λ) = aλ+ 12 bλ2

+

∫(0,∞)

(e−λx− 1+ λx1x<1

)π (dx) , (57)

where a ∈ R, b ∈ [0,∞) and π is a Borel measure on (0,∞) such that∫

(0,∞)(1∧x2)π (dx) <∞.A result similar to the theorem below was proved in [13, Theorem 4.2] in the context of Bernoullileaf percolation. The proof in the present framework is very similar, but since it demonstrates thecrucial appearance of the branching mechanism ψ , we include a brief account of it here.

Theorem 3.12. Let ξ be an infinitely extensible offspring distribution. Then, there exists afunction ψ : [0,∞)→ R of the form (57) such that ψ ′(1) = 1 and

ϕξ (r ) = r + ψ(1− r ) , r ∈ [0, 1] .

Conversely, suppose that ψ is of the Levy–Khintchine form (57) and suppose that there existsλ0 ∈ (0,∞) such that ψ(λ0) > 0. Then, the function

r ∈ [0, 1] ↦−→ r +ψ(λ0(1− r ))λ0ψ ′(λ0)

(58)

is the generating function of an infinitely extensible offspring distribution.

Proof. We first assume that ξ is an infinitely extensible offspring distribution. Let (ξn, αn),n ∈ N, be as in Definition 3.11. This implies that ϕξ can be extended analytically to the intervalIn = ( −αn

1−αn, 1). Moreover for all r ∈ In , and all k ≥ 2,

ϕ(k)ξ (r ) =

(1− αn)k−1

1− ϕ′ξn (αn)ϕ

(k)ξn(αn + (1− αn)r)

is positive. Since limn→∞αn = 1, it makes sense to set ψ(λ) = ϕξ (1 − λ) − (1 − λ), forall λ ∈ [0,∞). Then, ψ is C∞ on (0,∞) and (−1)kψ (k)(λ) ≥ 0, for all λ ∈ (0,∞) andall k ≥ 2. We then apply to ψ (2) Bernstein’s theorem on completely monotone functions (seee.g. Feller [17, Theorem XIII.7.2]): there exists a Radon measure ν on (0,∞) and b ≥ 0 such thatψ (2)(λ) = b+

∫(0,∞) e−λxν(dx). Observe that ψ(0) = 0 and ψ ′(1) = 1− ϕ′ξ (0) = 1− ξ (1) = 1.

Then, we set π (dx) = x−2ν(dx) and a = 1−b+∫

(0,∞)(e−x−1x<1)xπ (dx). This easily entails

that ψ is of the form (57). For the converse result, differentiation shows that (58) defines anoffspring distribution. Let α ∈ [0, 1) and λ := λ0/(1 − α). Then a straightforward calculationshows that ϕξλ (r ) = r + ψ(λ(1 − r ))/λψ ′(λ) is such that ξ (α)

λ = ξ . Hence ξ is infinitelyextensible.

3.2.2. Definition of growth processes. Characterisation of their laws

Definition 3.13 (Growth Processes). Let (Ω ,G,P) be a probability space. For all λ ∈ [0,∞), letFλ : Ω → T be a GW(ξ ∗λ , c∗λ;µ

∗

λ)-real forest as in Definition 2.14, for an offspring distributionξ ∗λ , and edge parameter c∗λ and an initial distribution µ∗λ. We say that (Fλ)λ∈[0,∞) is a growthprocess if for all λ′ ≥ λ, there exists a hereditary property Aλ,λ′ ⊂ T such that P-a.s. for allλ′ ≥ λ, Fλ = RAλ,λ′ (Fλ′ ).



Remark 3.14. Note that if (Fλ)λ∈[0,∞) is a growth process, then by (48), P-a.s. for any λ′ ≥ λ,Fλ ⪯ Fλ′ .

Remark 3.15. Suppose that the family (Fλ)λ∈[0,∞) satisfies the weaker consistency property thatfor all λ′ ≥ λ, P-a.s. Fλ = RAλ,λ′ (Fλ′ ), then by (47), we easily construct a modification of theprocess that is a growth process according to Definition 3.13.

This definition is meaningful without the assumption of Galton–Watson marginal distribu-tions, and the two remarks also still hold in this higher generality.

We next introduce specific one-parameter families of infinitely extensible offspring distribu-tions that play a key role. To that end, we fix the following setting:

− a branching mechanism ψ : [0,∞)→ R of the form (57), where we furthermore assumethe following:

allow deaths : ∃λ0 ∈ (0,∞) : ψ(λ0) > 0,

conservative :∫

0+

dr(ψ(r ))−

= ∞ ;(59)

− a probability measure ϱ on [0,∞) that is distinct from δ0.

The fact that ψ is a convex function and that ψ(0) = 0 has several consequences. First, observethat ψ has either one or two roots: we denote by q the largest one, and since ψ takes positivevalues eventually, we obtain (q,∞) = λ ∈ [0,∞) : ψ(λ) > 0. Moreover, ψ ′(0+) exists in[−∞,∞).

− If ψ ′(0+) < 0, we say that ψ is super-critical (and q > 0, necessarily).− If ψ ′(0+) = 0, we say that ψ is critical (and q = 0, necessarily).− If ψ ′(0+) > 0, we say that ψ is sub-critical (and q = 0, necessarily).

Next, for all λ ∈ [q,∞), we introduce two probability measures on N that are denoted by ξλ andµλ and we also define cλ ∈ (0,∞), as follows:

cλ = ψ ′(λ) , ϕξλ (r ) = r +ψ ((1− r )λ)λψ ′(λ)

and

ϕµλ (r ) =∫

[0,∞)e−λy(1−r )ϱ(dy) , r ∈ [0, 1].

(60)

More explicitly, we get ξλ(0) = ψ(λ)λψ ′(λ) , ξλ(1) = 0, and for all k ≥ 2,

ξλ(k) =λk−1

ψ (k)(λ)

k!ψ ′(λ)=

λk−1

k!ψ ′(λ)

(b1k=2 +

∫(0,∞)

xke−λx π (dx)). (61)

Recall (27): we easily check that ξλ is conservative iff the second assumption in (59) holds true.The following theorem provides a useful classification of growth processes.

Theorem 3.16. Let (Fλ)λ∈[0,∞) be a growth process as in Definition 3.13. By (48), P-a.s. λ ↦→Z+0 (Fλ) is non-decreasing and we set Z = limλ→∞Z+0 (Fλ). Then, only the two following casesoccur.

(I) If P(Z < ∞) = 1, there exists a proper and conservative offspring distribution ξ , thereexists c ∈ (0,∞) and there exists a probability measure µ distinct from δ0, such that

∀ k ∈ N , limλ→∞

ξ ∗λ (k) = ξ (k) , limλ→∞

µ∗λ(k) = µ(k) and limλ→∞

c∗λ = c. (62)



Moreover, there exists a unique non-increasing function α : [0,∞)→ Dξ such that

limλ→∞

αλ = 0 and ∀λ ∈ [0,∞) , ξ ∗λ = ξ(αλ) , c∗λ = c(αλ) and µ∗λ = µ

αλ ,

where for all α ∈ Dξ , (ξ (α), c(α), µα) is defined by (54).(II) If P(Z = ∞) > 0, there exists a triplet (ψ, ϱ, β) that satisfies the following.

(i) ψ is a function of the Levy–Khintchine form (57) that satisfies (59).(ii) ϱ is a probability measure [0,∞) distinct from δ0,

(iii) β : [0,∞)→ [q,∞) is a non-decreasing function such that limλ→∞β(λ) = ∞. Here,q stands for the largest root of ψ .

(iv) For all λ ∈ [0,∞),

ξ ∗λ = ξβ(λ) , c∗λ = cβ(λ) and µ∗λ = µβ(λ)

where for all λ ∈ [q,∞), (ξλ, cλ, µλ) is defined by (60).

Moreover, if another triplet (ψ∗, ϱ∗, β∗) satisfies (i)–(iv), there exists a constant κ ∈ (0,∞)such that ϱ∗(dx) = ϱ(dx/κ), and for all λ ∈ [0,∞), ψ∗(λ) = ψ(κλ)/κ and β∗(λ) =β(λ)/κ .

Proof. Let Tλ be a GW(ξ ∗λ , c∗λ)-real tree: its law is then Qξ∗λ ,c∗λ. We keep the same notation as in

Definition 3.13. For all λ′ ≥ λ, we recall from (22) the definition of the hereditary property A−λ,λ′

and we set

αλ,λ′ := P(Tλ′ ∈ A−

λ,λ′

)= Qξ∗

λ′,c∗λ′

(T \ A−

λ,λ′

).

By Theorem 2.19, ((ξ ∗λ′

)(αλ,λ′ ), (c∗λ′

)(αλ,λ′ ), (µ∗λ′

)αλ,λ′ ) = (ξ ∗λ , c∗λ, µ∗

λ). This first implies thatλ ↦→ c∗λ is non-decreasing. Recall from the definition of growth processes that µ∗0(0) < 1. Thisentails that µ∗λ(0) < 1, for all λ ∈ [0,∞), because ϕµ∗λ (0) ≤ ϕµ∗λ (α0,λ) = ϕµ∗0 (0) = µ∗0(0) < 1.Next, (55) easily entails

ϕξ∗λ′

(r ) = r +c∗λc∗λ′

·

((1− αλ,λ′ )ϕξ∗λ

(r−αλ,λ′

1−αλ,λ′

)− (r − αλ,λ′ )

)and

ϕµ∗λ′

(r ) = ϕµ∗λ

(r−αλ,λ′

1−αλ,λ′

).

(63)

Moreover, by (56), (1 − αλ,λ′ )(1 − αλ′,λ′′ ) = 1 − αλ,λ′′ , for all λ′′ ≥ λ′ ≥ λ. This impliesthat αλ,λ′ is non-decreasing in λ′ and non-increasing in λ. Then, for all λ ∈ [0,∞), we setαλ = limλ′→∞αλ,λ′ .• We first assume that P(Z < ∞) = 1. We denote by µ the law of Z . Since µ∗λ is the law of

Z+0 (Fλ), by definition, we get limλ→∞µ∗

λ(k) = µ(k), for all k ∈ N.We next claim that for all λ ∈ [0,∞), αλ < 1. We argue by contradiction: let us suppose that

αλ = 1. Then, for all λ′′ ≥ λ′ ≥ λ and all r ∈ [0, 1], we get

ϕµ∗λ (r ) = ϕµ∗λ′′

(αλ,λ′′ (1− r )+ r

)≥ ϕµ∗

λ′′

(αλ,λ′ (1− r )+ r

).

But limλ′′→∞ϕµ∗λ′′

(αλ,λ′ (1− r )+ r

)= ϕµ(αλ,λ′ (1−r )+r ) and limλ′→∞ϕµ

(αλ,λ′ (1− r )+ r

)=

ϕµ(1) = 1, since we have supposed αλ = 1. This implies that µ∗λ(0) = 1, which is impossible asalready proved. Note that this argument also entails that µ(0) < 1.

An elementary compactness argument shows that there is a sub-probability measure ξ on Nand a sequence λn ∈ [λ,∞), n ∈ N, increasing to ∞, such that limn→∞ξ

∗

λn(k) = ξ (k), for all



k ∈ N. This convergence entails limn→∞ϕξ∗λn(r ) = ϕξ (r ), for all r ∈ [0, 1). Next observe that

since αλ < 1, (63) implies that ϕξ∗λ can be extended analytically to the interval ( −αλ1−αλ, 1). Thus,

for all 0 < r < 1, in (63), we can let λ′→∞ (along the sequence λn) and there are two cases toconsider: since λ′ ↦→ c∗

λ′is non-decreasing, if c := limλ′→∞c∗

λ′<∞, then we get

ϕξ (r ) = r +c∗λc·

((1− αλ)ϕξ∗λ

(r − αλ1− αλ

)− (r − αλ)

), (64)

and if limλ′→∞c∗λ′= ∞, then we get ϕξ (r ) = r . However, the last possibility would imply that

ξ (1) = 1, which is impossible because ξ is the limit of proper offspring distributions. Thus (64)holds true. Note that (64) entails that ϕξ (1) = 1 and that ξ is a proper offspring distribution.Moreover, ξ is completely determined by (64). This implies (62). Also note that (64) easilyentails that αλ ∈ Dξ and that (ξ (αλ), c(αλ), µαλ) = (ξ ∗λ , c∗λ, µ

∗

λ).The function λ ∈ [0,∞) ↦→ αλ ∈ Dξ is clearly non-increasing. Let us prove uniqueness:

suppose that ϕµ∗λ (r ) = ϕµ(αλ + (1− αλ)r ) = ϕµ(γλ + (1− γλ)r ). Since µ(0) < 1, ϕµ is strictlyincreasing, and we get αλ = γλ. We next set α = limλ→∞αλ. Recall that ϕµ∗λ (0) = ϕµ(αλ) ≥ϕµ(α), which implies ϕµ(0) ≥ ϕµ(α) as λ → ∞. Thus, α = 0, since ϕµ is strictly increasing.This completes the proof of Case (I).

• We now assume that P(Z = ∞) > 0. We claim that α1 = limλ→∞α1,λ = 1. We argue bycontradiction: let us suppose that α1 < 1. We fix r < 1. Since µ∗1 = (µ∗λ)

α1,λ for all λ ∈ (1,∞),we get

ϕµ∗1(r ) = E

[(α1,λ + (1− α1,λ)r )Z+0 (Fλ)

]≤ E

[(α1 + (1− α1)r )Z+0 (Fλ)

]−→λ→∞

E[(α1 + (1− α1)r )Z 1Z<∞

].

We now let r go to 1 and we get 1 = ϕµ∗1 (1) ≤ P(Z < ∞), which contradicts the assumptionP(Z = ∞) > 0.

Since limλ→∞α1,λ = 1 and since ξ ∗1 = (ξ ∗λ )(α1,λ), ξ ∗1 is infinitely extensible. Theorem 3.12implies that there exists ψ1 of the form (57) that satisfies ψ ′1(1) = 1 and ϕξ∗1 (r ) = r +ψ1(1− r ),r ∈ [0, 1]. We denote by q the largest root of ψ1. Since ψ1(1) = ξ ∗1 (0) ≥ 0, we get q ≤ 1. Then,observe that since ξ ∗1 is conservative, ψ1 satisfies

∫0+ du/(ψ1(u))− = ∞. Thus, ψ1 satisfies (59).

We next prove that µ∗1 is a mixture of Poisson distributions: we set g(r ) = ϕµ∗1(1 − r )

and observe that g(r ) = ϕµ∗λ (1 − (1 − α1,λ)r ), for all λ ∈ [1,∞). Since limλ→∞α1,λ = 1,g can be analytically extended to (0,∞), and we easily see that g is completely monotone.Bernstein’s theorem entails that there exists a probability measure ϱ on [0,∞) such thatϕµ∗1

(r ) =∫

e−y(1−r )ϱ(dy), for all r ∈ [0, 1]. Note that ϱ(0) = µ∗1(0) < 1. Thus ϱ is distinctfrom δ0.

We now set ψ = c∗1ψ1 and we define β : [0,∞)→ [q,∞) by setting

β(λ) = 1− αλ,1, if λ ∈ [0, 1), β(1) = 1, and

β(λ) =1

1− α1,λ, if λ ∈ (1,∞).

Recall notation (ξλ, cλ, µλ) from (60). Then, for all λ ∈ [0,∞), we check that ξ ∗λ = ξβ(λ),c∗λ = cβ(λ) and µ∗λ = µβ(λ) by applying (54) to α = αλ,1 and (ξ, c, µ) = (ξ ∗1 , c∗1, µ

∗

1) if λ < 1,and by applying (55) to α = α1,λ and (ξ, c, µ) = (ξ ∗λ , c∗λ, µ

∗

λ), if λ > 1. We have proved that(ψ, ϱ, β) satisfies (i)–(iv). The last point of the theorem is not difficult to check: we leave it tothe reader.



3.3. Convergence to Lévy forests

In this section we prove under a necessary and sufficient condition that every growth processconverges almost surely to a Levy forest represented as a T-valued random variable. We firstprove convergence in law for the branching processes and by the general criterion in Theorem 3.9we deduce the desired result.

Continuous-state branching process. Continuous-state branching processes (CSBPs for short)are the continuous analogue, in time and space, of Galton–Watson branching processes. Theywere introduced by Jirina [27] and Lamperti [31,29,30]. We first recall standard results on CSBPswhose proof can be found in Silverstein [40] and Bingham [6].

We fix a branching mechanism ψ : [0,∞) → R of the form (57) and we assume thatψ satisfies (59). We also fix a probability measure ϱ on [0,∞) that is distinct from δ0. A[0,∞)-valued Feller process Zϱ = (Zϱa )a∈[0,∞) defined on the probability space (Ω ,G,P) isa continuous-state branching process with branching mechanism ψ and initial distribution ϱ (aCSBP(ψ, ϱ) for short) if its transition kernels are characterised by the following:

E[

e−θ Zϱb+a

Zϱb]= exp

(−Zϱb u(a, θ)

), a, b, θ ∈ [0,∞) ,

where for all θ ∈ [0,∞), the function a ↦→ u(a, θ) is the solution of the differential equation∂u(a, θ)/∂a = −ψ(u(a, θ)), with u(0, θ) = θ . Under our assumptions on ψ , the differentialequation satisfied by u(·, θ) has a unique nonnegative solution that is defined on [0,∞) for allθ ∈ [0,∞). In particular, note that the null function is the unique solution for θ = 0.

Let us rewrite the equation characterising u in a more convenient way. Recall that q standsfor the largest root of ψ . If θ ∈ 0, q, then u(a, θ) = θ for all a ∈ [0,∞). Observe that if θ > q(resp. θ < q) then a ↦→ u(a, θ) is a decreasing (resp. increasing) function converging to q as atends to infinity. An easy change of variables then entails

∀ a ∈ [0,∞) , ∀ θ ∈ [0,∞) \ 0, q ,∫ θ

u(a,θ )

drψ(r )

= a. (65)

We derive from the equation governing u and from basic properties of Laplace transforms thatfor all a ∈ [0,∞), Zϱa is integrable iff both ψ ′(0+) and

∫[0,∞) yϱ(dy) are finite quantities. In this

case we get

E[Zϱa ] = e−ψ′(0+)a

∫[0,∞)

yϱ(dy). (66)

We next easily derive from (65) that

P(

lima→∞

Zϱa = 0)+ P

(lim

a→∞Zϱa = ∞

)= 1 and

P(

lima→∞

Zϱa = ∞)= 1−

∫[0,∞)

e−qxϱ(dx) .

Thus, P(lima→∞Zϱa = ∞

)> 0 iff q > 0. Namely, this happens only in the super-critical cases.

It is easy to derive from (65) that

P(

lima→∞

Zϱa = 0)= P

(∃a ≥ 0 : Zϱa = 0

)⇐⇒

∫∞ drψ(r )

<∞. (67)



That is why condition (67) is called the finite time extinction assumption. Now observe that forall a ∈ (0,∞), θ ↦→ u(a, θ) is an increasing function and under (67),

v(a) := limθ→∞

u(a, θ) exists and satisfies∫∞

v(a)

drψ(r )

= a , a ∈ (0,∞). (68)

Observe that the function v : (0,∞)→ (q,∞) is decreasing and one-to-one. We next define theextinction time of Zϱ by

E(Zϱ) = infa ∈ (0,∞) : Zϱa = 0

, (69)

with the convention inf∅ = ∞. We then easily get

P ( E(Zϱ) ≤ a ) =∫

[0,∞)ϱ(dy) exp (−v(a)y) , a ∈ (0,∞). (70)

We refer to [6] for more details on CSBPs.

Convergence in law to Lévy forests. Let us fix a branching mechanism ψ of the form (57) thatfurthermore satisfies (59). We also fix ϱ, a probability measure on [0,∞) that is distinct from δ0.Recall that q stands for the largest root of ψ . For all λ ∈ [q,∞), recall from (60) the definitionof (ξλ, cλ, µλ).

We denote by Fλ : Ω → T a GW(ξλ, cλ;µλ)-real forest. We do not need to assume that it isa growth process: here, we are only interested in the convergence in law of these random treeswhen λ goes to infinity. We also denote by Tλ : Ω → T, a GW(ξλ, cλ)-real tree. Namely the lawof Tλ is Qξλ,cλ and the law of Fλ is Pξλ,cλ,µλ , as in Definition 2.14.

We first consider the corresponding branching processes. As an easy consequence of (30) and(65), for all a, θ ∈ [0,∞), we get

wλ(a, θ) := E[exp(−θ Z+a (Tλ))

]= Qξλ,cλ

[exp(−θ Z+a )

]= 1−

u(a, λ(1− e−θ )

)λ

(71)

and (33) also implies

E[exp(−θ Z+a (Fλ))

]= Pξλ,cλ,µλ

[exp(−θ Z+a )

]=

∫[0,∞)

exp(−y u(a, λ(1− e−θ ))

)ϱ(dy). (72)

This easily implies that for all a ∈ [0,∞), 1λ

Z+a (Fλ) → Zϱa in distribution on [0,∞). We nextuse a result due to Helland [24, Theorem 6.1] that shows that the following convergence actuallyholds in distribution in the space of cadlag functions D([0,∞),R) equipped with Skorokhod’smetric:(

1λ

Z+a (Fλ))

a∈[0,∞)−→

(Zϱa)

a∈[0,∞) as λ→∞. (73)

In the following lemma, we compute the law of the h-leaf-length erased forest Rh(Fλ).

Lemma 3.17. Let ψ be a function of the form (57) that satisfies (59). Let ϱ be a probabilitymeasure on [0,∞), distinct from δ0. Let Fλ be as above. Then, the following holds true.

(i) For all h ∈ (0,∞), all a ∈ [0,∞) and all λ ∈ [q,∞),

Qξλ,cλ (Γ ≤ h) = 1−u(h, λ)λ

and Rh(Fλ)(law)= Fu(h,λ). (74)



Moreover,

E[exp(−θ Z (h)

a (Fλ))]=

∫[0,∞)

ϱ(dy) exp(−y u

(a, u(h, λ)(1− e−θ )

)). (75)

(ii) The laws Pξλ,cλ,µλ of the GW-real forests Fλ are tight on T as λ → ∞ if and only if ψsatisfies∫

∞ drψ(r )

<∞ .

(iii) Assume that∫∞ dr/ψ(r ) <∞ and recall from (68) the definition of v. Then,(

Z (h)a (Fλ) ; Rh(Fλ)

)−→

(Z+a (Fv(h)) ; Fv(h)

)weakly on N× T as λ→∞. Consequently,

limλ→∞

E[exp(−θ Z (h)

a (Fλ))]=

∫[0,∞)

ϱ(dy) exp(−y u

(a, v(h)(1− e−θ )

)). (76)

Proof. Recall that Ah = Γ ≥ h is a hereditary property on T. Thus, Rh(Tλ) = RAh (Tλ).We then see that α := Qξλ,cλ (T \ A−h ) = Qξλ,cλ (Γ ≤ h). Then, recall that Qξλ,cλ (Γ ≤ h) =limθ→∞wλ(h, θ). Thus, (71) entails the equality in (74). Then, Theorem 2.19 entails that RAh (Fλ)is a GW(ξ (α)

λ , c(α)λ ;µ

αλ )-real forest, and we easily see that

(ξ (α)λ , c(α)

λ , µαλ ) = (ξu(h,λ), cu(h,λ), µu(h,λ)) ,

which proves the point (i).We next prove (ii). Theorem 3.3 asserts that the laws of the Fλ are tight on T as λ → ∞

iff for each a, h ∈ (0,∞) the laws of the Z (h)a (Fλ), λ ∈ [q,∞), are tight on N as λ → ∞.

If∫∞ dr/ψ(r ) = ∞, (65) entails that limλ→∞u(h, λ) = ∞, and since ϱ = δ0, (75) implies

that the laws of the Z (h)a (Fλ) are not tight on N as λ → ∞. If

∫∞dr/ψ(r ) < ∞, then

limλ→∞u(h, λ) = v(h) < ∞, and (75) implies (76), which proves that the laws of the Z (h)a (Fλ)

are tight on N, and the proof of (ii) is complete.We next observe limλ→∞(ξu(h,λ), cu(h,λ), µu(h,λ)) = (ξv(h), cv(h), µv(h)). Then, Lemma 2.17

applies, which entails (iii).

We can explore (ii) further, as follows: if we consider the special case when Fλ and Fλ0 arerelated by leaf-length erasure Rh(λ0,λ)Fλ = Fλ0 for some h(λ0, λ) > 0, for all λ > λ0, we seefrom (74) that u(h(λ0, λ), λ) = λ0 and hence, by (65),

h(λ0, λ) =∫ λ

u(h(λ0,λ),λ)

drψ(r )

=

∫ λ

λ0

drψ(r )

−→

∫∞

λ0

drψ(r )

=: h(λ0,∞),

as λ→∞.

We will show in Theorem 3.18(ii) that (Fλ)λ≥0 converges to some limit F in (T, δ), if h(λ0,∞) <∞, and that then Rh(λ0,∞)F = Fλ0 . If h(λ0,∞) = ∞, there is no limit in (T, δ), so we mightwant to search for bigger spaces where the increasing limit of Fλ can be taken. However, withh(λ0,∞) = ∞ we would not be able to recover Fλ0 from any limiting metric space F byleaf-length erasure, so this search seems doomed.

Theorem 3.18. Let ϱ be a Borel probability measure on [0,∞) distinct from δ0. Let ψ :[0,∞) → R be a branching mechanism of Levy–Khintchine form (57). We assume that ψ



satisfies (59) and (67). We denote by q largest root. Let (Ω ,G,P) be a probability space. Forall λ ∈ [q,∞), let Fλ : Ω → T be a GW(ξλ, cλ;µλ)-real forest where (ξλ, cλ, µλ) is given by(60). Then the following joint convergence holds true in distribution in D([0,∞),R)× T:( (

1λ

Z+a (Fλ))

a∈[0,∞); Fλ

)−→λ→∞

((Zϱa )a∈[0,∞) ; F

), (77)

where the limit is as follows.

(i) The limiting forest F is a (ψ, ϱ)-Levy forest. We denote by Pψ,ϱ its distribution on T whichonly depends on ψ and ϱ. Furthermore, we have Pψ∗,ϱ∗ = Pψ,ϱ iff there exists κ ∈ (0,∞)such that ψ∗(λ) = ψ(κλ)/κ and ϱ∗(dy) = ϱ(dy/κ).

(ii) Rh(F) is a GW(ξv(h), cv(h);µv(h))-real forest.(iii) The process Zϱ is a CSBP(ψ, ϱ).(iv) limh→0

1v(h) Z (h)

a (F) = Zϱa , in probability, for all a ∈ [0,∞).

In the present paper, the limit of Fλ serves as an implicit definition of Levy forests. Inthis, we follow [13, Section 5.1], which is consistent with earlier work [33,11,12], whichoriginally defined Levy forests in terms of properties of height functions. Let us also mentionthe construction of Abraham and Delmas [2] by truncation of super-critical Levy trees and by aGirsanov formula.

Proof. By Lemma 3.17(iii) and Corollary 3.4, limλ→∞Fλ = F weakly in T and (ii) holds true.Moreover, the last point of (i) is then a consequence of Theorem 3.16.

We next prove the joint convergence. To simplify notation, we set Y λ= ( 1

λZ+a (Fλ))a∈[0,∞).

Recall from (73) that Y λ converges weakly in D([0,∞),R) to a CSBP(ψ, ϱ). Thus, the jointlaws of the (Y λ, Fλ) are tight in D([0,∞),R) × T as λ→ ∞. We want to prove that there is aunique limiting distribution by proving (iv) for a possible limit. To that end, let us assume thatalong a sequence (λn)n∈N increasing to ∞, the following convergence holds in distribution inD([0,∞),R)× T.(

Y λn ; Fλn

)−→n→∞

(Y ; F

). (78)

By a slight abuse of notation, we assume that Y and F are defined on (Ω ,G,P). We fixa ∈ [0,∞) and h ∈ (0,∞). We know from Lemma 3.17 that Z (h)

a (Fλn ) converges in law on N.Then the laws of (Z (h)

a (Fλn ); Y λn ; Fλn ) are tight on N×D([0,∞),R)×T. There is an increasingsequence of integers (n(k))k∈N such that the following convergence holds in distribution inN× D([0,∞),R)× T.(

Z (h)a (Fλn(k) ) ; Y λn(k) ; Fλn(k)

)−→k→∞

(X ; Y ′ ; F ′

). (79)

Again, to simplify notation, we assume that X , Y ′ and F ′ are defined on (Ω ,G,P). Clearly(Y ′, F ′) and (Y, F) have the same law. The δ-continuity of Rh implies that limk→∞(Rh(Fλn(k) );Fλn(k) ) = (Rh(F ′), F ′) weakly on T2. Moreover, F ′ is a (ψ, ϱ)-Levy forest and by (ii), Rh(F ′)has the same law as Fv(h). Lemma 3.17 asserts that, weakly on N × T, limk→∞(Z (h)

a (Fλn(k) );Rh(Fλn(k) )) = (Z+a (Fv(h)); Fv(h)), which has the same law as (Z+a (Rh(F ′)); Rh(F ′)). Thus(X; Rh(F ′)) has the same law as (Z+a (Rh(F ′)); Rh(F ′)), which implies that

P-a.s. X = Z+a (Rh(F ′)) = Z (h)a (F ′) , (80)



the last equality being the definition of Z (h)a . We next recall from (74) that Qξλ,cλ (Γ >

h) = λ−1u(h, λ). We recall from (32) that conditionally given Z+a (Fλ), Abv (a, Fλ) isdistributed according to Q⊛Z+a (Fλ)

ξλ,cλ . Since Z (h)a (Fλ) = ⟨M0(Abv (a, Fλ)), 1Γ>h⟩, we know that

conditionally given Z+a (Fλ), the variable Z (h)a (Fλ) has a binomial law with parameters λ−1u(h, λ)

and Z+a (Fλ). Thus, for all θ ∈ [0,∞) and all continuous bounded functions f , we easily get

E[

e−θ Z (h)a (Fλn(k) ) f (λ−1

n(k) Z+a (Fλn(k) ))]

= E

[(1−

u(h, λn(k))λn(k)

(1− e−θ

))Z+a (Fλn(k) )

f (λ−1n(k) Z+a (Fλn(k) ))

].

We then pass to the limit as k →∞ to get

E[exp(−θX ) f (Y ′a)

]= E

[exp

(−v(h)Y ′a

(1− e−θ

))f (Y ′a)

].

This entails that conditionally given Y ′a , X is a Poisson random variable with parameter v(h)Y ′a .By (80), for all K , ε ∈ (0,∞), we get

P( 1v(h)

Z (h)a (F)− Ya

> ε

)= P

( 1v(h)

Z (h)a (F ′)− Y ′a

> ε

)≤

Kε2v(h)

+ P(Y ′a > K ).

This entails that for all a ∈ [0,∞), limh→01v(h) Z (h)

a (F) = Ya in probability. This entails theuniqueness of the limit for the joint laws, which completes the proof of the theorem.

Almost sure convergence of growth processes. We turn now to the main result of this section,which asserts almost sure convergence for growth processes and their branching processes. Weonly consider the most interesting case when the limiting tree is not a GW-tree.

Theorem 3.19. Let (Ω ,G,P) be a probability space and let Fλ : Ω → T, λ ∈ [0,∞), be agrowth process such that P(Z+0 (Fλ)→∞) > 0. Let (ψ, ϱ, β) be the triplet governing the growthprocess as specified in Theorem 3.16(II). We furthermore assume

∫∞dr/ψ(r ) < ∞. Then there

exists a random CLCR real tree F : Ω → T such that

P-a.s. limλ→∞

δ(Fλ, F

)= 0. (81)

The random tree F is a (ψ, ϱ)-Levy forest as defined in Theorem 3.18 and there exists a cadlagCSBP(ψ, ϱ) denoted by (Zϱa )a∈[0,∞) such that for all a ∈ [0,∞)

P-a.s. Zϱa = limλ→∞

1β(λ)

Z+a (Fλ) = limh→0+

1v(h)

Z (h)a (F). (82)

Moreover, the same limits hold in L1 if ψ ′(0+) and∫

[0,∞) yϱ(dy) are both finite.

We expect this convergence to hold a.s. uniformly in a, but we have been unable to prove this.Standard martingale methods do not appear to apply here.

Proof. We keep the notation of Theorem 3.16: Fλ is a GW(ξβ(λ), cβ(λ);µβ(λ))-real forest, wherefor all λ ∈ [q,∞), (ξλ, cλ, µλ) is given by (60). Lemma 3.17 asserts that for all a ∈ [0,∞)and all h ∈ (0,∞), Z (h)

a (Fλ) converges in law as λ goes to∞. By Remark 3.14, Theorem 3.9applies and there exists F such that (81) holds. By Theorem 3.18, F is a (ψ, ϱ)-Levy forestand there exists a cadlag CSBP(ψ, ϱ) denoted by (Zϱa )a∈[0,∞) such that for all a ∈ [0,∞),



limh→0+1v(h) Z (h)

a (Fλ) = Zϱa in probability and such that the following convergence holds weaklyon D([0,∞),R)× T( (

1β(λ)

Z+a (Fλ))

a∈[0,∞); Fλ

)−→λ→∞

((Zϱa )a∈[0,∞) ; F

). (83)

We fix a ∈ [0,∞) and we turn now to the a.s. convergence of the branching processes. Weuse the following martingale argument. For all λ ∈ [0,∞), we denote by Hλ the sigma-fieldgenerated by the variables Z+a (Fλ′ ), λ′ ∈ [λ,∞). We then claim that ( 1

β(λ) Z+a (Fλ))λ∈[0,∞) is anonnegative backward martingale with respect to (Hλ)λ∈[0,∞).

Proof of the claim. We first fix λ, λ′ ∈ [0,∞), such that λ′ > λ, and we denote by Tλ′ aGW(ξβ(λ′), cβ(λ′))-real tree. We use the notation of Definition 3.13 and we denote by Aλ,λ′ ⊆ Tthe family of hereditary properties under which the growth process is consistent. We then set

αλ,λ′ := Qξβ(λ′),cβ(λ′)

(T \ A−

λ,λ′

)= Qξβ(λ′),cβ(λ′)

(T \ A−

λ,λ′

).

The Aλ,λ′ -reduced offspring distribution is ξ(αλ,λ′ )β(λ′) = ξβ(λ), and a brief computation involving

(35) and (60) entails

αλ,λ′ = 1−β(λ)β(λ′)

. (84)

Let G : T→ [0,∞) be measurable and let n′ ≥ n. We next compute

B := E[G(Abv (a, Fλ)

)1Z+a (Fλ′ )=n′ ; Z+a (Fλ)=n

].

We recall from (32) that conditionally given the event Z+a (Fλ′ ) = n′, Abv (a, Fλ′ ) isdistributed as a forest of n′ independent GW(ξβ(λ′), cβ(λ′))-real trees. By (24) in Lemma 2.11,RAλ,λ′ (Abv (a, Fλ′ )) = Abv (a, Fλ). Then, by Theorem 2.19, conditionally given the eventZ+a (Fλ′ ) = n′, Abv (a, Fλ) is a GW(ξβ(λ), cβ(λ);µ)-real forest where µ stands for the binomialdistribution with parameters n′ and 1− αλ,λ′ . This implies the following:

B = P(Z+a (Fλ′ ) = n′

)Pξβ(λ),cβ(λ),µ

[1Z+0 =nG

]= P

(Z+a (Fλ′ ) = n′

) (n′

n

)(1− αλ,λ′ )

nαn′−nλ,λ′

Q⊛nξβ(λ),cβ(λ)

[G]

=P(Z+a (Fλ′ ) = n′

)P(Z+a (Fλ) = n

) (n′

n

)(1− αλ,λ′ )

nαn′−nλ,λ′

E[G(Abv (a, Fλ)

)1Z+a (Fλ)=n

].

We next fix the real numbers λp > · · · > λ0 ≥ 0, and the integers n p ≥ · · · ≥ n1 ≥ 0, andfor all n0 ∈ 0, . . . , n1, we set

u(n0) = P(Z+a (Fλp ) = n p ; . . . ; Z+a (Fλ1 ) = n1 ; Z+a (Fλ0 ) = n0

).

We now apply the previous computation to G(Abv (a, Fλp−1 ))=1Z+a (Fλp−1 )=n p−1 ; ... ; Z+a (Fλ0 )=n0

,λ′ = λp, n′ = n p, λ = λp−1 and n = n p−1 to get

u(n0) =P(

Z+a (Fλp ) = n p)

P(

Z+a (Fλp−1 ) = n p−1

) (n p

n p−1

)(1− αλp−1,λp

)n p−1αn p−n p−1λp−1,λp

×

P(

Z+a (Fλp−1 ) = n p−1 ; . . . ; Z+a (Fλ0 ) = n0

).



This entails u(n0) = P(Z+a (Fλp ) = n p)∏

1≤k≤p

( nknk−1

)(1− αλk−1,λk

)nk−1αnk−nk−1λk−1,λk

. Thus we get

u(n0) =(

n1

n0

)(1− αλ0,λ1

)n0αn1−n0λ0,λ1

P(Z+a (Fλp ) = n p ; . . . ; Z+a (Fλ1 ) = n1

). (85)

We now fix λ′ > λ and we denote by Pλ′ the set of events of the form

Z+a (Fλp ) ≥ n p ; . . . ; Z+a (Fλ1 ) ≥ n1, λ1, . . . , λp ∈ [λ′,∞) ,

n1, . . . , n p ∈ N , p ∈ N∗ .

Clearly Pλ′ contains Ω , it is stable under intersection and it generates Hλ1 . Moreover, (85) withλ = λ0 and λ1 = λ

′, easily entails that for all n ∈ N, and all A ∈ Pλ′ ,

E[1Z+a (Fλ)=n1A

]= E

[(Z+a (Fλ′ )

n

)(1− αλ,λ′ )

nαZ+a (Fλ′ )−nλ,λ′

1A

].

A monotone class argument entails that the same equality holds for all A ∈ Hλ′ . Thus, for allA ∈ Hλ′

E[Z+a (Fλ)1A

]=

∑n≥0

E

[n(

Z+a (Fλ′ )n

)(1− αλ,λ′ )

nαZ+a (Fλ′ )−nλ,λ′

1A

]

= E[(1− αλ,λ′ )Z+a (Fλ′ )1A

]=β(λ)β(λ′)

E[Z+a (Fλ′ )1A

],

by (84), which immediately implies the claim.

The theorem of almost sure convergence of nonnegative backward martingale implies that forevery sequence (λn)n∈N that increases to∞, P-a.s. limn→∞

1β(λn ) Z+a (Fλn ) exists. Then, the joint

convergence (83) also implies that the limiting r.v. is necessarily P-a.s. equal to Zϱa .By (66), if both ψ ′(0+) and

∫[0,∞) yϱ(dy) are finite, then Zϱa is integrable. Standard results on

backward martingales then entail that for all λ ∈ [0,∞), 1β(λ) Z+a (Fλ) is integrable and that

limλ→∞

E[ 1

β(λ) Z+a (Fλ)− Zϱa ] = 0 .

Since β may have jumps, additional arguments are required to get the first equality in(82), which is proved so far only along a subsequence. To simplify notation, we first setXλ =

1β(λ) Z+a (Fλ). Then, we fix ε ∈ (0,∞) and λ∗ ∈ (0,∞) such that β(λ∗) > 0. We

recursively define an increasing sequence (λn)n∈N that tends to ∞ by fixing λ0 > λ∗ andby setting λn+1 = infλ > λn : logβ(λ) > ε + logβ(λn+). Then observe that for allλ, λ′ ∈ (λn, λn+1), e−ε ≤ β(λ′)/β(λ) ≤ eε. Since the growth process is ⪯-non-decreasing,(48) entails that P-a.s. for all n ∈ N, for all λ, λ′, λ′′ ∈ (λn, λn+1) such that λ′′ ≤ λ ≤ λ′,

e−εXλ′′ ≤ Xλ ≤ eεXλ′ .

For all λ ∈ (0,∞) we also define the sigma-field Hλ− =⋂λ′<λHλ′ and we also denote by Hλ+

the sigma-field generated by⋃λ′>λHλ. We next fix a sequence (h p)p∈N that decreases to 0. By

standard arguments for nonnegative martingales and for nonnegative backward martingales, weget the following: P-a.s. for all n ∈ N,

E[Xλ∗ |Hλn−] = limp→∞

Xλn−h p =: Xλn− and

E[Xλ∗ |Hλn+] = limp→∞

Xλn+h p =: Xλn+ .



Now observe that (Xλn−)n∈N and (Xλn+)n∈N are positive backward martingales with respect to thebackward filtrations (Hλn−)n∈N and (Hλn+)n∈N. So, they both converge P-a.s. By Theorem 3.9,Fλ−h p converges P-a.s., as p →∞. We denote the limit by Fλ−. By Lemma 2.17, observe that(Xλ−h p , Fλ−h p ) converges in distribution to a GW(ξβ(λ−), cβ(λ−), µβ(λ−))-real forest as p →∞.Moreover, Theorem 3.18 applies to such laws, which implies a joint weak convergence thatis similar to (83) with the left-limit process instead of the normal one. This easily entails thatP-a.s. limn→∞Xλn− = Zϱa . A similar argument also implies that P-a.s. limn→∞Xλn+ = Zϱa .

This proves that for all ε ∈ (0,∞), there exists an event Ωε ∈ F of probability one on whichthe following holds true: Xλn , Xλn+ and Xλn− tend to Zϱa as n→∞, and

∀n ∈ N, ∀λ ∈ [λn, λn+1],e−ε min(Xλn+, Xλn , Xλn+1 ) ≤ Xλ ≤ eε max(Xλn+1−, Xλn , Xλn+1 ).

Thus, on Ωε, e−εZϱa ≤ lim infλ→∞Xλ ≤ lim supλ→∞Xλ ≤ eεZϱa . This easily entails the firstequality in (82).

It remains to prove the assertions concerning Z (h)a (F). To that end, recall from (68) the

definition of the function v that is continuous decreasing from (0,∞) to (q,∞). We denoteby v−1 its inverse and for all λ ∈ [0,∞), we set F ′λ = Rv−1(q+1+λ)(F). It clearly defines a growthprocess that δ-converges to F . This process is governed by a triplet of the form (ψ, ϱ, β ′) withβ ′(λ) = q + 1+ λ. Now observe that for all a ∈ [0,∞) and all h such that v−1(1+ q) > h > 0,

1v(h)

Z (h)a (F) =

1β ′(v(h)− 1− q)

Z+a(F ′v(h)−1−q

).

We now apply the first equality in (82) to that specific growth process to obtain the secondequality in (82).

As an application of the previous results on growth processes, we first state a simplecharacterisation of Levy forests that is used to derive limit theorems of GW-forests to Levyforests.

Theorem 3.20. Let (Ω ,G,P) be a probability space. Let F : Ω → T be a random CLCR realtree. We assume that P(F = Υ ) > 0 and that for all sufficiently small h ∈ (0,∞), Rh(F) is aGW-real forest. Then, the following holds true.

(a) Either P(Z+0 (F) <∞) = 1 and F is a GW-real forest .(b) Or P(Z+0 (F) = ∞) > 0 and F is a Levy forest as in Theorem 3.18.

Proof. First note that P(Rh(F) = Υ ) > 0, for all sufficiently small h. Let h0 be such that for allh ∈ (0, h0), Rh(F) is a GW-real forest such that P(Rh(F) = Υ ) > 0. Then, for all λ ∈ [0,∞),we set Fλ = Rh0e−λ (F). Clearly, (Fλ)λ∈[0,∞) is a growth process. We then apply Theorem 3.16.Namely, since Z+0 (Fλ) → Z+0 (F), either P(Z+0 (F) < ∞) = 1 and the growth process is as inTheorem 3.16 (I): by Lemma 2.17, Fλ converges in law to a GW-forest as λ→∞, which impliesthat F is a GW-real forest. Or P(Z+0 (F) = ∞) > 0, and the growth process is as in Theorem 3.16(Il): the growth process is then governed by a triplet (ψ, ϱ, β). Since the laws of the Fλ are tightin T as λ→∞, Lemma 3.17(ii) implies that

∫∞dr/ψ(r ) <∞, and Theorem 3.19 implies that

F is a Levy forest.

Theorem 3.20 allows us to strengthen Lemma 2.16, as follows. Recall from Section 2.3 thedefinition of the sigma-field Ba+.



Theorem 3.21. Let Q be a Borel probability measure on T. We first assume that Q(0 < D <

∞) > 0. We also assume that for all a ∈ [0,∞), Q(Z+a < ∞) = 1. Then, the followingassertions are equivalent.

(i) For every a ∈ [0,∞), the conditional distribution given Z+a of Abv (a, ·) under Q is Q⊛Z+a .(ii) For every a ∈ [0,∞), the conditional distribution given Ba+ of Abv (a, ·) under Q is

Q⊛Z+a .(iii) Q is the distribution of a GW(ξ, c)-real tree, for a certain c ∈ (0,∞) and a certain proper

conservative offspring distribution ξ .

Proof. If we assume (iii), then a ↦→ Z+a is cadlag Q-a.s., hence Lemma 2.16 applies andwe get (ii) and (i). It remains to prove (i) ⇒ (iii) without assuming, as in Lemma 2.16, thata ↦→ Z+a is cadlag Q-a.s. To this end, first note that (i) implies Q(Z+0 = 1) = 1, as in theproof of Lemma 2.16 in Appendix C.2. Next, observe that Γ ≥ D on T \ Υ , and recall fromLemma 2.5 that limh→0 D Rh = D. Thus, there exists h0 ∈ (0,∞) such that for all h ∈ (0, h0),Q(0 < D Rh < ∞) > 0 and qh := Q(Γ > h) > 0. Then it makes sense to define Qh as thelaw of Rh under Q( · |Γ > h) and we have proved that Qh(0 < D <∞) > 0. Next, observe thata ↦→ Z+a Rh = Z (h)

a+ is cadlag Q-a.s., which implies that a ↦→ Z+a is cadlag Qh-a.s.Let us show that Qh satisfies (i). Let us fix a ∈ [0,∞), n ∈ N \ 0 and a measurable function

G : T → [0,∞). The property (i) for Q, (24) and arguments similar to those used in the proofof Theorem 2.18 imply the following

Qh

[1Z+a =nG(Abv (a, ·))

]= q−1

h Q[1Z+a Rh=nG(Abv (a, Rh))

]= q−1

h

∑N≥n

Q[1Z+a =N ; Z+a Rh=nG (Rh(Abv (a, ·)))

]= q−1

h

∑N≥n

Q[1Z+a =N Q

⊛N[1Z+a Rh=nG(Rh)

]]= q−1

h

∑N≥n

Q(Z+a = N )(

Nn

)(1− qh)N−nqn

h Q⊛nh [G]

= Qh(Z+a = n)Q⊛nh [G] .

This implies that Qh satisfies (i). Then Lemma 2.16 implies that Qh is the law of a GW-real tree.Thus, for any h ∈ (0, h0), Rh under Q is the law of a GW-real forest and since Q(Z+0 = 1) = 1,Theorem 3.20 entails that Q is the law of a GW-real tree, which completes the proof.

3.4. Invariance principles for GW-trees

In this section we apply the tightness results of Section 3.1 and the previous results on Levyforests to obtain limit theorems for discrete GW-real trees. We consider two asymptotic regimes:we first discuss convergence in distribution to GW-real trees and we next discuss convergence toLevy forests.

Notation. Unless the contrary is explicitly mentioned, all the random variables are defined onthe same probability space (Ω ,G,P). For all p ∈ N,

– ξ p= (ξ p(k))k∈N is an offspring distribution such that ξ p(1) < 1,

– ν p= (ν p(k))k∈Z is given by ν p(k) = ξ p(k + 1) if k ≥ −1 and ν p(k) = 0 if k < −1,



– µp= (µp(k))k∈N is a probability distribution on N such that µp(0) < 1,

– (Y pk )k∈N is a discrete-time GW-Markov chain with initial distribution µp and offspring

distribution ξ p.

Discrete GW-real trees and forests. We modify the lifetime part of Definition 2.14 and say that aT-valued random variable Tp is called a discrete GW(ξ p)-real tree if its distribution Q satisfies

Q[1k=nG(ϑ)g(D)

]= ξ p(n)Q⊛n[G]g(1) .

This is just a way to view graph trees as metric spaces by joining the vertices by intervals of unitlength. Now let Tp(n), n ≥ 1, be independent copies of Tp. Let Np be an N-valued variable withlaw µp, independent of (Tp(n))n≥1. We then set

Fp := ⊛1≤n≤Np Tp(n) ,

with the convention that Fp = Υ if Np = 0. Then, Fp is a random forest of Np independentdiscrete GW(ξ p)-real trees.Tree-scaling. The main purpose of this section is to obtain convergence in law of Fp whensuitably rescaled. More precisely, let T ∈ T and c ∈ (0,∞); let (T, d, ρ) be a representative ofT . Then, (T, cd, ρ) is a CLCR real tree and its pointed isometry class only depends on T and c:we denote it by cT . Note that the function (c, T ) ↦→ cT is continuous.Notation. The symbol ∗ stands for the convolution product of measures on R. For every measurem on R, we set m∗1 = m and for all n ≥ 1, we set m∗(n+1)

= m ∗ m∗n . Also, ⌊·⌋ stands for theinteger-part function.

Convergence to Galton–Watson trees. We first state a convergence result for GW-Markov chainsthat are rescaled in time but not in space. This result is quite close to Grimvall’s [21, Theorem3.4] that is actually a limit-theorem for GW-chains that are rescaled in time and space. Sinceit is not explicitly written in [21], we state it here as a lemma: its proof can be adapted in astraightforward way from that of [21, Theorem 3.4] that strongly relies on [20, Theorem 2.2′](whose proof also extends to our setting).

Lemma 3.22. Let (γp)p∈N be a positive sequence converging to ∞. Then, the followingstatements are equivalent.

(i) There exist a probability measure ν on Z with ν(0) < 1 and a probability measure µ on Nwith µ(0) < 1, such that the following two limits hold in law as p→∞.

(ν p)∗⌊γp⌋ −→ ν and µp−→ µ.

(ii) The one-dimensional marginal distributions of (Y p⌊γpa⌋)a∈[0,∞) converge to those of an

N-valued process that is not constant.(iii) The process (Y p

⌊γpa⌋)a∈[0,∞) converges weakly on D([0,∞),R) to a continuous-time

N-valued Galton–Watson branching process that is not constant.

Remark 3.23. Assume that (i), (ii) and (iii) hold true. Note that ν has to be an infinitely divisibledistribution on Z. Then ν = δ0 is the law of X1, where (X t )t∈[0,∞) is a compound Poissonprocess with holding-time parameter c and jump law π , a probability measure on Z\0. Denoteby (X p

k )k∈N a random walk with jump law ν p, and initial state X p0 = 0. Standard arguments

imply that (X p⌊γp t⌋)t∈[0,∞) converges weakly on D([0,∞),R) to (X t )t∈[0,∞). Since we deal with

integer-valued processes, the joint law of the first jump-time and the size of the jump of X p



converges to that of X . The assumptions on ν p first imply that the support of π is included in−1, 1, 2, . . .. If we set ξ (k) = π (k − 1), k ∈ N, then it is easy to see that ξ is the (properand conservative) offspring distribution of the continuous-time N-valued GW-branching processmentioned in (iii) and that c is its lifetime parameter. The previous joint convergence then entailsthat for all k ∈ N,

limp→∞

ξp

# (k) = ξ (k) , limp→∞

µp(k) = µ(k) , limp→∞

γp(1− ξ p(1)) = c , (86)

where ξ p# (k) = ξ p(k)/(1− ξ p(1)) if k = 1 and ξ p

# (1) = 0.

The previous result is used to derive the following limit theorem. Recall that Fp stands for arandom forest of Np independent discrete GW(ξ p)-real trees and that Np has law µp.

Theorem 3.24. Let (γp)p∈N be a positive sequence converging to∞. We assume that the one-dimensional marginal distributions of (Z+a ( 1

γpFp))a∈[0,∞) converge to those of an N-valued

process that is not constant. Then, there exists a GW(ξ, c;µ)-real forest F , as in Definition 2.14with µ(0) < 1, such that((

Z+a

(1γp

Fp

))a∈[0,∞)

;1γp

Fp

)(law)−→p→∞

((Z+a (F)

)a∈[0,∞) ; F

),

weakly on D([0,∞),R)× T. Moreover, (86) holds true.

Proof. First observe that Y pk := Z+kγp

( 1γpFp), k ∈ N, is a GW(ξ p)-Markov chain with initial

distribution µp. Thus, Lemma 3.22(i), (ii) and (iii) hold true.We first prove convergence for single GW(ξ p)-discrete trees Tp. This corresponds to the case

where µp(1) = 1, to which Lemma 3.22 also applies. Thus, (Z+a ( 1γpTp))a∈[0,∞) converges weakly

on D([0,∞),R) as p → ∞ to a continuous-time GW(ξ, c)-branching process whose initialvalue is 1. Moreover, ξ is proper and conservative and (86) holds true. Since continuous-timeGW-branching processes have no fixed-time discontinuity, convergence of finite-dimensionalmarginals holds true. This entails the tightness of the laws of 1

γpTp, p ∈ N, since for all

a, h ∈ [0,∞), Z (h)a ( 1

γpTp) ≤ Z+a ( 1

γpTp), and by Theorem 3.3.

We next want to prove that the real trees 1γpTp converge weakly to a GW(ξ, c)-real tree by

showing that every weak limit satisfies the branching property of Theorem 3.21. To that end,observe first that the joint laws of (Z+a ( 1

γpTp))a∈[0,∞) and 1

γpTp, are tight on D([0,∞),R) × T.

Let (Ya)a∈[0,∞) and T be such that((Z+a

(1γp(k)

Tp(k)

))a∈[0,∞)

;1γp(k)

Tp(k)

)−→k→∞

((Ya)a∈[0,∞) ; T

)(87)

weakly on D([0,∞),R)×T along the increasing sequence of positive integers (p(k))k∈N. Withoutloss of generality, by the Skorokhod representation theorem (and by a slight abuse of notation),we can assume that (87) holds almost surely. To simplify notation we set Tk =

1γp(k)

Tp(k). Thus,we assume that((

Z+a (Tk))

a∈[0,∞) ; Tk

)−→k→∞

((Ya)a∈[0,∞) ; T

)a.s. in D([0,∞),R)× T. (88)

We first claim that

P(Z+0 (T ) = 1

)= 1 and P

(0 < D(T ) <∞

)= 1. (89)



Proof of (89). We first use the following standard result on Skorokhod convergence for N-valuedcadlag functions: by (88), the first jump time and the value at the first jump time of the processesZ+·

(Tk) converge a.s. to the first jump time and the value at the first jump time of the process Y .Observe that since Z+0 (Tk) = 1, we get

D(Tk) = infa ∈ [0,∞) : Z+a (Tk) = 1 and k(Tk) = Z+D(Tk )

(Tk) .

Then, if we set D′ = infa ∈ [0,∞) : Ya = 1 and k′ = YD′ , the previous arguments and thecontinuity of Abv entail that

D′ = limk→∞

D(Tk) , k′ = limk→∞

k(Tk) and Abv (D′, T ) = δ– limk→∞

ϑ(Tk). (90)

Since Y is a GW(ξ, c)-branching process, D′ and k′ are independent, D′ is exponentiallydistributed with mean 1/c and k′ has law ξ that is proper and conservative. Thus, 0 < D′ < ∞and k′ = 1 a.s.

Let (Tk, dk, ρk) be any representative of Tk and let (T , d, ρ) be any representative of T . Then,P-a.s. for any fixed r ∈ (0, D′), for all sufficiently large k the closed ball BTk (ρk, r ) rooted at ρk

is equivalent to the interval [0, r ] rooted at 0 and since δ(Tk, T )→ 0, the closed ball BT (ρ, r )rooted at ρ is also equivalent to the interval [0, r ] rooted at 0, which implies that Z+0 (T ) = 1 andD(T ) ≥ r . This proves that

P-a.s. D(T ) ≥ D′ > 0 and Z+0 (T ) = 1. (91)

It remains to prove that D(T ) < ∞ a.s., and by (91), this boils down to proving that T is notthe pointed isometry class of a half-line rooted at its finite end. If k′ = 0, then for any r > D′,(90) implies that for all sufficiently large k, k(Tk) = 0, which implies ϑ(Tk) = Υ and thusAbv (D′, T ) = Υ . Consequently, if k′ = 0, T is not the pointed isometry class of a half-linerooted at its finite end, and therefore D(T ) <∞.

Since k′ = 1 a.s., it only remains to consider the case where k′ ≥ 2. To that end, we firstprove that

P-a.s. on k′ ≥ 2, lim supk→∞

D(ϑTk) > 0. (92)

Indeed, first note that for any ε ∈ (0,∞), 1lim supk→∞D(ϑTk )<ε ;k′≥2 ≤ lim infk→∞

1D(ϑTk )≤ε ;k(Tk )≥2. This inequality combined with Fatou lemma entails

P(lim sup

k→∞D(ϑTk) < ε ; k′ ≥ 2

)≤ lim inf

k→∞P(D(ϑTk) ≤ ε ; k(Tk) ≥ 2

).

Now observe that P(D(ϑTk) > ε ; k(Tk) ≥ 2

)= E

[P(D(Tk) > ε)k(Tk )1k(Tk )≥2

], by the

branching property for discrete Galton–Watson trees. This, combined with (90) entails that

limk→∞

P(D(ϑTk) > ε ; k(Tk) ≥ 2

)=

∑n≥2

(e−cε)n

ξ (n) −−−−→ε→0

1− ξ (0)− ξ (1)

= limk→∞

P(k(Tk) ≥ 2

).

Thus, limε→0limk→∞P(D(ϑTk) ≤ ε ; k(Tk) ≥ 2

)= 0, which implies (92).

Then, denote by (T ′, d ′, ρ ′) a representative of Abv (D′, T ). By (92), a.s. if k′ ≥ 2, thereexists r > 0 such that D(ϑTk) > r for infinitely many k, and (90) implies that the closed ballBT ′ (ρ ′, r ) is equivalent to k′ copies of [0, r ] glued at 0. It implies that T is not a half-line andthus, D(T ) <∞. This completes the proof of the claim (89).



We next denote by Q the law of T . Then, (89) can be rephrased as

Q(Z+0 = 1) = 1 and Q( 0 < D <∞ ) = 1. (93)

Let us also denote by Q p(k) the law of Tk . We fix a ∈ [0,∞) and for all k ∈ N, we setak := ⌊γp(k)a⌋/γp(k) → a, as k → ∞. Let G1,G2 : T → [0,∞) be continuous and bounded.First observe that

E[G1(Blw (ak, Tk)

)G2(Abv (ak, Tk)

)]= E

[G1(Blw (ak, Tk)

)Q

⊛Z+ak (Tk )p(k) [G2]

],

by the branching property for discrete Galton–Watson trees. As k →∞, (88) and the continuityof Blw and Abv stated in Lemma 2.3 imply that

Q [G1(Blw (a, ·))G2(Abv (a, ·))] = E[G1(Blw (a, T )

)G2(Abv (a, T )

)]= E

[G1(Blw (a, T ))Q⊛Ya [G2]

]. (94)

This equality extends to all nonnegative measurable functions G1 and G2. Since Q(Z+0 = 1) = 1,(94) with G1 ≡ 1 and G2(·) = f (Z+0 (·)) first implies Q[ f (Z+a )] = E[ f (Ya)]. This first provesthat Q(Z+a < ∞) = 1 for any a ∈ [0,∞), and it also entails for any measurable functionsG : T→ [0,∞) and f : N→ [0,∞) that

Q[

f (Z+a )G(Abv (a , · ))]= E

[f (Ya)Q⊛Ya [G]

]= Q

[f (Z+a )Q⊛Z+a [G]

].

Therefore, the conditional distribution given Z+a of Abv (a , · ) under Q is Q⊛Z+a .We thus have proved that Theorem 3.21 applies to Q that is therefore the law of a GW(ξ ′, c′)-

real tree. Since Ya under P has the same law as Z+a under Q, we easily get (ξ ′, c′) = (ξ, c).Since T is a GW(ξ, c)-real tree, a ↦→ Z+a (T ) is cadlag and it has no fixed discontinuity. Thus,

for any a ∈ (0,∞), this process is left-continuous at time a a.s., and Z+a (T ) is equal a.s. to ameasurable functional G1 of Blw (a, T ). The previous arguments and (94) entail that for anya ∈ [0,∞), Z+a (F) = Ya a.s., and since both processes are cadlag, we get Z+

·(T ) = Y a.s. This

proves the uniqueness of the limit in the joint convergence and it actually proves the theorem inthe case of single trees, namely when µp

= δ1, p ∈ N.

Let us prove the general case. Denote by Q p the law of 1γpTp. Let F : D([0,∞),R)×T→ R

be bounded and continuous for the product topology. We proved that∫

F(Z+·

(T ); T )Q p(dT )−→

∫F(Z+

·(T ); T )Qξ,c(dT ). Now observe that for any k ∈ N,∫

TF(Z+·

(T ) ; T)

Q⊛kp (dT )

=

∫Tk

F(Z+·

(T1)+ · · · + Z+·

(Tk) ; T1 ⊛ · · ·⊛ Tk)

Q⊗kp (dT1 . . . dTk)

Since k independent continuous-time GW-branching processes have distinct jump times a.s., weget

limp→∞

∫Tk

F(Z+·

(T1)+ · · · + Z+·

(Tk) ; T1 ⊛ · · ·⊛ Tk)

Q⊗kp (dT1 . . . dTk)

=

∫Tk

F(Z+·

(T1)+ · · · + Z+·

(Tk) ; T1 ⊛ · · ·⊛ Tk)

× Q⊗kξ,c(dT1 . . . dTk)

=

∫T

F(Z+·

(T ) ; T)

Q⊛kξ,c(dT )



This implies the following:

limp→∞

E[

F(

Z+·

(1γpFp

);

1γpFp

)]= lim

p→∞

∑k∈N

µp(k)∫T

F(Z+·

(T ) ; T)

Q⊛kp (dT )

=

∑k∈N

µ(k)∫T

F(Z+·

(T ) ; T)

Q⊛kξ,c(dT ) ,

which completes the proof of the theorem.

Convergence to Lévy forests. We consider now the convergence of GW-trees to Levy forests. Inthese cases, the profiles of the trees are rescaled in time and space. More precisely, we make thetwo following assumptions.

(A1) There is a positive sequence (γp)p∈N converging to ∞, such that the process( 1

p Y p⌊γpa⌋)a∈[0,∞) converges weakly on D([0,∞),R) to a CSBP(ψ, ϱ), where ϱ(0) < 1

and∫

0+dr

(ψ(r ))−= ∞.

(A2) Set Ep = infk ∈ N : Y (p)k = 0 and denote by E the extinction time of a CSBP(ψ, ϱ). We

assume that∫∞ drψ(r ) <∞ and that 1

γpEp −→ E , weakly on [0,∞], as p→∞.

Grimvall [21, Theorem 3.4] asserts that (A1) is equivalent to the following weak convergenceon R:

ν p(·

p

)∗p⌊γp⌋

−→p→∞

ν and µp(·

p

)−→p→∞

ϱ ,

where ν is an infinitely divisible spectrally positive law such that∫R e−θxν(dx) = eψ(θ ) . Analytic

necessary and sufficient conditions equivalent to such a convergence can be found for instancein [25, Theorem II.3.2].

If we assume (A1) and∫∞ drψ(r ) <∞, then we can show that (A2) is equivalent to the following

lim infp→∞

(ϕ⌊γp⌋ξ p (0)

)p> 0 ,

where ϕnξ p stands for the nth iterate of ϕξ p . We leave the details to the reader (see also thecomments following [11, Theorem 2.3.1]). Let us mention that (A1) implies (A2) when ξ p

= ξ ,for all p ∈ N. In this case, ψ is necessarily a γ -stable branching mechanism, namely ψ(λ) = λγ ,γ ∈ (1, 2] (see [11, Theorem 2.3.2]).

We now state the main result of the section. To that end, recall from (65) and (68) the notationu(a, θ) and v(a), recall from Theorem 3.18 the definition of Levy forests and recall that Fpstands for a random forest of Np independent discrete GW(ξ p)-real trees, where Np has law µp.

Theorem 3.25. Assume (A1) and (A2). Then, there exist a (ψ, ϱ)-Levy forest F and aCSBP(ψ, ϱ) denoted by (Zϱa )a∈[0,∞) such that for all a ∈ [0,∞), P-a.s. limh→0+

1v(h) Z (h)

a (F) =Zϱa , and such that weakly on D([0,∞),R)× T,((

1p

Z+a

(1γp

Fp

))a∈[0,∞)

;1γp

Fp

)(law)−→p→∞

((Zϱa )a∈[0,∞) ; F

). (95)

Remark 3.26. We first mention that a closely related result has been proved by quite differentmethods in [11, Theorem 2.3.1 and Corollary 2.5.1]: this result only deals with critical or sub-critical GW-trees, however the convergence holds for the contour process, which is a stronger



convergence. Note that in the super-critical cases, the contour process is not a well-suitedapproach.

Remark 3.27. As noticed in [11], (A1) and (A2) are in some sense the minimal assumptionsunder which the convergence (95) holds. Indeed, (A1) and

∫∞ drψ(r ) < ∞, do not necessarily

imply (A2): see [11, pp. 60–61] for a counterexample. Note that (ψ, ϱ)-Levy forests can only bedefined as locally compact real trees if

∫∞ drψ(r ) <∞. Moreover, if we assume (A1),

∫∞ drψ(r ) <∞

and 1γpFp → F , then, by the δ-continuity of Γ , the total heights converge too, which implies

(A2).

Proof. Recall that Tp stands for a single discrete GW(ξ p)-real tree and observe that Z+kγp( 1γpTp),

is a discrete-time GW(ξ p)-Markov chain whose initial state is equal to 1. For all a, θ, h ∈ [0,∞)and all p ∈ N, we set

u p(a, θ) = −p log E[exp(−1pθ Z+a (

1γp

Tp))] and vp(h) = −p log P(Γ (1γp

Tp) < h ) .

Next, observe that Y pk := Z+kγp

( 1γpFp), k ∈ N, is a discrete-time GW(ξ p)-Markov chain with

initial distribution µp, to which (A1) and (A2) apply. Then, by (A1),

ϕµp (exp(−1p

u p(a, θ))) = E[

exp(−1pθY (p)⌊γpa⌋)

]−→p→∞

∫[0,∞)

ϱ(dy) e−yu(a,θ )

and by (A2) and (70), we get

ϕµp (exp(−1pvp(h))) = P(

1γp

Ep < h ) −→p→∞

∫[0,∞)

ϱ(dy) e−yv(h) .

We next use the following basic result (known as the second theorem of Dini).

(R) Let f p : [0,∞) → [0,∞), p ∈ N, be a sequence of monotonic functions convergingpointwise to a continuous function f . Then the convergence is uniform on every compactinterval of [0,∞).

We first note that the functions f p(θ ) = ϕµp (e−θ/p) and f (θ ) =∫

[0,∞) ϱ(dy)e−θy are strictlymonotonic and continuous, so that the inverses f −1

p converge to f −1 pointwise on the interval(ϱ(0) , 1]. Hence, for all a, θ ∈ [0,∞) and for all h ∈ (0,∞), we get u p(a, θ) → u(a, θ)and vp(h) → v(h). Next observe that u p is monotone in each component and that vp is non-increasing. Thus, by (R),

(ap, θp, h p) −→p→∞

(a, θ, h) ∈ [0,∞)2× (0,∞)

H⇒ u p(ap, θp) −→p→∞

u(a, θ) and vp(h p) −→p→∞

v(h). (96)

We next fix h ∈ (0,∞) and we set Fhp := R⌊γph⌋(Fp). It is easy to see that Fh

p is a forest ofdiscrete GW-real trees as defined at the beginning of the section. We want to apply Theorem 3.24to Fh

p . To that end, first denote by ξ ph its offspring distribution and by µp

h the law of the numberof trees in this forest. We do not need to compute them explicitly to see that ξ p

h (1) < 1 andµ

ph (0) < 1. Next fix a ∈ [0,∞) and observe that conditionally given Y p

⌊γpa⌋ = Z+⌊γpa⌋(Fp) = n,

the law of Z+⌊γpa⌋(Fh

p ) is binomial with parameters n and P(Γ (Tp) ≥ ⌊γph⌋). To simplifynotation, we set h p = ⌊γph⌋/γp, which tends to h as p→∞, and we define φp(h, θ) ∈ [0,∞)



by

exp(−1pφp(h, θ)) = P

(Γ (Tp) < ⌊γph⌋

)+ e−θP

(Γ (Tp) ≥ ⌊γph⌋

)= 1− (1− e−θ )(1− exp(−

1pvp(h p))).

By (96), φp(h, θ)→ v(h)(1−e−θ ) and E[exp(−θ Z+a ( 1γpFh

p ))]−→∫

[0,∞) ϱ(dy)e−yu(a,v(h)(1−e−θ ) ).Theorem 3.24 applies and the following convergence holds weakly on D([0,∞),R)× T:(

(Z+a (1γp

Fhp ))a∈[0,∞);

1γp

Fhp

)−→p→∞

((Z+a (Fh))a∈[0,∞); Fh) .

Here, it is easy to see that Fh is a GW(ξv(h), cv(h);µv(h))-real forest, where we recall that theone-parameter family of laws (ξλ, cλ, µλ) is derived from ψ and ϱ by (60).

Now observe that δ(Rh( 1γpFp), 1

γpFh

p ) ≤ 1γp

. Moreover, for each fixed a, there exists arandom variable ∆p : Ω → N such that ∆p ≥ Z+a ( 1

γpFh

p ) − Z (h)a ( 1

γpFp) ≥ 0, and such

that conditionally given Z+⌊γpa⌋(

1γpFp) = n, the law of ∆p is binomial with parameters n and

P(h p < Γ ( 1

γpTp) ≤ h p+

2γp

), which tends to 0 as p→∞, by (96). Since the laws of the random

variables Z+⌊γpa⌋(

1γpFh

p ) are tight, we get ∆p → 0, in probability. Thus, for all h ∈ (0,∞) and alla ∈ [0,∞), (Z (h)

a ( 1γpFp) ; Rh( 1

γpFp)) converges to (Z+a (Fh) ; Fh) weakly on N×T, as p→∞.

Let F be a (ψ, ϱ)-Levy forest. Theorem 3.18 (ii) asserts that Rh(F) and Fh have the samelaw for all h ∈ (0,∞). Then, by Corollary 3.4, 1

γpFp → F weakly on T as p →∞. Thus, we

have proved

1γp

Fp(law)−−−→

p→∞F and

(Z (h)

a (1γp

Fp) ; Rh( 1γp

Fp)) (law)−−−→

p→∞

(Z (h)

a (F) ; Rh(F)).

(97)

Then, note that the laws of ( ( 1p Z+a ( 1

γpFp))a∈[0,∞) ;

1γpFp), p ∈ N, are tight on D([0,∞),R)×T.

Let (p(k))k∈N be an increasing sequence of integers such that((1

p(k)Z+a

(1γp(k)

Fp(k)

))b∈[0,∞)

;1γp(k)

Fp(k)

)(law)−−−→

k→∞

((Wb)b∈[0,∞) ; F

)(98)

where (Wb)b∈[0,∞) is a CSBP(ψ, ϱ). By Theorem 3.19, the proof of the joint convergence will becomplete if we show that for each a ∈ [0,∞), Z (h)

a+(F)→ Wa in probability as h → 0. To thatend, first note that by (97), there exists an increasing sequence of integers (kℓ)ℓ∈N such that thefollowing limit holds true weakly on N2

× T(Z (h)

a+

(1

γp(kℓ)Fp(kℓ)

);

1p(kℓ)

Z+a

(1

γp(kℓ)Fp(kℓ)

);

1γp(kℓ)

Fp(kℓ)

)−→ℓ→∞

(X ;Wa ; F

). (99)

The δ-continuity of Rh and (97) imply that (X, Rh(F)) and (Z (h)a (F), Rh(F)) have the same law,

which implies that X = Z (h)a (F) a.s.

Next, observe that the conditional law of Z (h)a ( 1

γpFp) given Z+a ( 1

γpFp) = n, is binomial with

parameters n and P(Γ (Tp) ≥ γph + γpa − ⌊γpa⌋). Previous computations and (99) imply that



for all θ ∈ [0,∞) and all bounded continuous functions f ,

E[e−θ Z (h)

a (F ) f (Wa)]= E

[e−Wav(h)(1−e−θ ) f (Wa)

].

Thus, the conditional law of Z (h)a (F) given Wa is a Poisson distribution with parameter v(h)Wa .

This implies that for all a ∈ [0,∞) and all h, K , ε ∈ (0,∞),

P( |Wa −1v(h)

Z (h)a (F)| > ε) ≤

Kε2v(h)

+ P(Wa > K ) ,

which implies the desired result.

Acknowledgements

We would like to thank an anonymous referee for carefully reading our manuscript and forgiving constructive comments that helped to improve the quality of the paper. This research wasin part supported by EPSRC grant GR/T26368/01 and ANR GRAAL Project ANR-14 CE25-0014.

Appendix A. Proofs of preliminary results on real trees.

Let us first recall basic results on the Gromov–Hausdorff metric (in the context of real trees).Let (T, d, ρ) and (T ′, d ′, ρ ′) be two CLCR real trees and let ε ∈ (0,∞). A function f : T → T ′

is a pointed ε-isometry if it satisfies the following conditions.

(a) f (ρ) = ρ ′.(b) dis( f ) := sup

|d(σ, s)− d ′( f (σ ), f (s))| ; σ, s ∈ T

< ε. This quantity is called the

distortion of f .(c) f (T ) is a ε-net of T ′. Namely, every point of T ′ is at distance at most ε of f (T ).

The following lemma is a translation into our tree context of [7, Corollary 7.3.28].

Lemma A.1. If δcpct(T, T ′) < ε, then there exists a pointed 4ε-isometry from T to T ′. If thereexists a pointed ε-isometry from T to T ′, then δcpct(T, T ′) < 4ε.

Recall (2) that gives the height of the branch point of two points σ, s ∈ T . If f : T → T ′ is apointed ε-isometry then, (2) implies that

∀σ, s ∈ T ,d(ρ, σ ∧ s)− d ′(ρ ′, f (σ ) ∧ f (s))

< 32ε. (100)

A.1. Proof of Lemma 2.3

Observe that for every CLCR real tree (T, d, ρ), Blw (a, T ) is compact and that the ball ofcentre ρ with radius r of Abv (a, T ) is equal to Abv (a, BT (ρ, a + r )). Thus, without loss ofgenerality, we only need to consider compact real trees.

For all a, b ∈ (0,∞), we easily see that δcpct (Blw (a, T ),Blw (b, T )) ≤ |a − b|. Assumethat (T ′, d ′, ρ ′) is a compact rooted real tree. The definition (4) of δcpct easily entails thatδcpct(Blw (a, T ),Blw (a, T ′)) ≤ 3δcpct(T, T ′) for all a ∈ (0,∞). Thus,

δcpct(Blw (a, T ),Blw (b, T ′)

)≤ |a − b| + 3δcpct(T, T ′) ,

which entails the joint continuity for Blw .



Next, define the following pseudo-metric da on T × T by

da(σ, s) = a ∨ d(ρ, σ )+ a ∨ d(ρ, s)− 2 (a ∨ d(ρ, σ ∧ s)) (101)

and say that σ ≡ s iff da(σ, s) = 0. Let ρa be the equivalence class of ρ. Then, (T/≡da , da, ρa)is isometric to Abv (a, T ). Note that 0 ≤ d(σ, s) − da(σ, s) ≤ 2a, which easily implies thatthe canonical projection from T to T/≡da is a 2a-pointed isometry and by Lemma A.1, we getδcpct(T,Abv (a, T )) ≤ 8a. Since Abv (a1 + a2, T ) = Abv (a1,Abv (a2, T )), we easily get

∀a, b ∈ [0,∞) , δcpct (Abv (a, T ),Abv (b, T )) ≤ 8|a − b|. (102)

Let (T ′, d ′, ρ ′) be a compact rooted real tree and let ε > δcpct(T, T ′). Lemma A.1 impliesthat there exists a 4ε-pointed isometry f from T to T ′ and (101) and (100) entail |da(σ, s) −d ′a( f (σ ), f (s))| ≤ 20ε, for all σ, s ∈ T . An easy argument shows that f induces a pointed28ε-isometry from Abv (a, T ) to Abv (a, T ′). By Lemma A.1 and (102) we get

∀a, b ∈ [0,∞) , δcpct(Abv (a, T ),Abv (b, T ′)

)≤ 112 δcpct(T, T ′)+ 8|a − b|,

which completes the proof of Lemma 2.3


Proof of Lemma 2.4(i). First note that for all a, h ∈ [0,∞) and every CLCR realtree T , Z+a (T ) = ⟨Ma(T )⟩ and Z (h)

a+(T ) = ⟨Ma(Rh(T ))⟩. Next, observe that Ma(T ) =M0(Abv (a, T )). So we only need to prove that M0 is measurable. The definition of δ entailsthat we only need to prove that the restriction of M0 to (Tcpct, δcpct) is measurable. To that end,we first prove the following claim.Claim 1. For all bounded Lipschitz functions F : Tcpct → [0,∞) and φ : [0,∞) → [0,∞),with φ vanishing in a neighbourhood of 0, and for all sufficiently small h ∈ (0,∞),

⟨(M0 Rh)( · ) , F · φ Γ ⟩ : Tcpct −→ [0,∞) is Borel-measurable. (Claim 1)

We first prove that (Claim 1) implies the desired result. For all T ∈ Tcpct, set G(T ) =F(T )φ(Γ (T )) and say that G is C-Lipschitz. Let h0 ∈ (0,∞), be such that φ(y) = 0, for ally ∈ (0, h0). Observe that for all h ∈ (0, h0/2),

∀T ∈ Tcpct ,⟨M0(Rh(T )),G⟩ − ⟨M0(T ),G⟩

≤ Ch #Z (h0/2)0 (T ) −→ 0

as h → 0.

By (Claim 1), ⟨M0(·),G⟩ is measurable and a monotone class argument shows that T ↦→⟨M0(T ), F · φ Γ ⟩ is measurable for all bounded measurable F : Tcpct → [0,∞) andfor all bounded Lipschitz φ : [0,∞) → [0,∞) vanishing in a neighbourhood of 0. Letφn : [0,∞) → [0,∞), n ∈ N, be a sequence of such functions such that φn ≤ φn+1 andsupn∈Nφn = 1(0,∞). By monotone convergence, ⟨M0( · ), F⟩ = limn→∞⟨M0( · ), F ·φn Γ ⟩, thatis therefore measurable. Thus, M0 is measurable and (Claim 1) entails Lemma 2.4 (i).

To prove (Claim 1), we prove (Claim 2) that is stated as follows. Fix h0 > h > 0. Fixφ : [0,∞) → [0,∞) a bounded Lipschitz function such that φ(y) = 0, for all y ∈ [0, h0]. FixF : Tcpct → [0,∞), a bounded Lipschitz function. Let (T, d, ρ) be a compact rooted real tree.Then, for all u ∈ [0, h), we set

Ψu(T ) = ⟨Mu (Rh(T )) , F · φ Γ ⟩ .



Denote by (T j , d, σ j ), j ∈ J , the subtrees of Rh(T ) above level u. Then Ψu(T ) =∑j∈J F(T j )φ(Γ (T j )). We then set

∆(T ) := d(ρ, σ ) ; σ ∈ Br(Rh(T )) ∪ Lf(Rh(T )) .

Note that ∆(T ) is a finite set. Then, we claim that

∀T ∈ Tcpct , ∀u ∈ (0, h) \∆(T ) , Ψu(T ′)→ Ψu(T )

as δcpct(T ′, T )→ 0. (Claim 2)

Let us first prove that (Claim 2) implies (Claim 1). To simplify notation, we write G = F ·φ Γand note that G is C-Lipschitz. Suppose that [u, v] ⊆ [0, h) \ ∆(T ). To each subtree T j abovelevel u in Rh(T ) corresponds a unique subtree of Rh(T ) above level v that is simply the treeT j shortened at its root by a line of length v − u. Thus |Ψu(T ) − Ψv(T )| ≤ C(v − u)# j ∈J : Γ (T j ) ≥ h0. This proves that u ↦→ Ψu(T ) is right-continuous on [0, h) \ ∆(T ). Forall K ∈ (0,∞) and for all u ∈ (0, h), we set Φu,K (T ) =

∫ 10 (K ∧ Ψuv(T ))dv. Since ∆(T )

is a finite set, it is Lebesgue negligible. Hence, (Claim 2) and dominated convergence implythat Φu,K : Tcpct → [0,∞) is δcpct-continuous. Dominated convergence also implies that forall T ∈ Tcpct, limu→0Φu,K (T ) = K ∧ Ψ0(T ). This entails that Ψ0 = ⟨(M0 Rh)( · ),G⟩ isBorel-measurable, which proves (Claim 1).

It remains to prove (Claim 2). We use the previous notation G. We fix u ∈ (0, h) \ ∆(T ).Since ∆(T ) is finite, we fix ε ∈ (0, u/4) that can be chosen arbitrarily small and such that[u − 2ε, u + 2ε] ⊆ (0, h) \ ∆(T ). Note that ε < h0/4. Let (T ′, d ′, ρ ′) be a compact rootedreal tree such that δcpct(Rh(T ), Rh(T ′)) < ε/4. By Lemma A.1 there exists a pointed ε-isometryf : Rh(T )→ Rh(T ′). We next denote by (T ′k , d ′, σ ′k), k ∈ J ′, the subtrees of Rh(T ′) above levelu, so that Ψu(T ′) =

∑k∈J ′G(T ′k ). Recall that (T j , d, σ j ), j ∈ J , stand for the subtrees of Rh(T )

above level u. We next set Jε = j ∈ J : Γ (T j ) > 2ε and we construct an injective functionπ : Jε → J ′ such that

∀ j ∈ Jε , δcpct(T j , T ′π ( j)

)< 76ε. (103)

Construction of π : for each j ∈ Jε, we fix γ j ∈ T j such that d(σ j , γ j ) = 2ε. Since d(ρ, σ j ) = uand σ j ∈ [[ρ, γ j ]], we get d(ρ, γ j ) = u + 2ε and d ′(ρ ′, f (γ j )) > u + ε, and there exists k ∈ J ′

such that f (γ j ) ∈ T ′k . We set π ( j) = k.π is injective: let i ∈ Jε \ j. Then γi ∧ γ j = σi ∧ σ j and (100) implies that

d ′(ρ ′, f (γi ) ∧ f (γ j )) < d(ρ, σi ∧ σ j ) + 3ε/2. Since [u − 2ε, u + 2ε] ⊆ (0, h) \ ∆(T ), weget d(ρ, σi ∧ σ j ) < u − 2ε. Consequently, d ′(ρ ′, f (γi ) ∧ f (γ j )) < u and π (i) = π ( j).

We next prove that

∀ j ∈ Jε , ∀γ ∈ T j such that d(σ j , γ ) ≥ 2ε , f (γ ) ∈ T ′π ( j). (104)

Indeed, since [u − 2ε, u + 2ε] ⊆ (0, h) \ ∆(T ), d(ρ, γ ∧ γ j ) > u + 2ε and (100) entailsd ′(ρ ′, f (γ ) ∧ f (γ j )) > u, which implies (104).

Next observe that for all j ∈ Jε,

d ′(σ ′π ( j), f (σ j )) ≤ d ′(σ ′π ( j), f (γ j ))+ d ′( f (γ j ), f (σ j ))

≤ d ′(ρ ′, f (γ j ))− u + d(σ j , γ j )+ ε

≤ d(ρ, γ j )+ ε − u + 2ε + ε = u + 2ε + ε − u + 2ε + ε = 6ε. (105)



We next define f j : T j → Rh(T ′) by setting f j (σ ) = f (σ ) if d(σ j , σ ) ≥ 2ε, and f (σ ) = σ ′π ( j)if d(σ j , σ ) < 2ε. We deduce from (104) that f j (T j ) ⊆ T ′π ( j). Next observe that if d(σ, σ j ) < 2ε,then (105) implies

d ′( f j (σ ), f (σ )) = d ′(σ ′π ( j), f (σ )) ≤ d ′(σ ′π ( j), f (σ j ))+ d ′( f (σ j ), f (σ ))

≤ 6ε + d(σ, σ j )+ ε < 9ε.

Hence, d ′( f j (σ ), f (σ )) < 9ε, for all σ ∈ T j . Consequently, dis( f j ) ≤ dis( f ) + 18ε < 19ε.We next prove that f j (T j ) is a 4ε-net of T ′π ( j): let σ ′ ∈ T ′π ( j) such that d ′(σ ′π ( j), σ

′) > 4ε.Since f is an ε-isometry, there exists σ ∈ Rh(T ) such that d ′( f (σ ), σ ′) < ε. Thus, d(ρ, σ ) >d ′(ρ ′, f (σ ))− ε > d ′(ρ ′, σ ′)− 2ε > u + 2ε, and (104) implies that σ ∈ T j and f j (σ ) = f (σ ),which proves that f j (T j ) is a 4ε-net of T ′π ( j). Thus, f j is a pointed 19ε-isometry from (T j , d, σ j )to (T ′π ( j), d ′, σ ′π ( j)), which entails (103) by Lemma A.1.

We next prove thatk ∈ J ′ : Γ (T ′k ) ≥ h0

⊆ π

(j ∈ J : Γ (T j ) ≥ h0/2

)⊆ π (Jε). (106)

Let σ ′ ∈ T ′k such that d ′(σ ′k, σ′) = h0. There exists σ ∈ Rh(T ) such that d ′( f (σ ), σ ′) < ε. We

then get

d(ρ, σ )− u > d ′(ρ ′, f (σ ))− ε − u > d ′(ρ ′, σ ′)− 2ε − u > h0/2 > 2ε.

Thus σ ∈ T j for a certain j ∈ Jε and f (σ ) ∈ T ′π ( j) by (104). Moreover, (2) easily entails thatd ′(ρ ′, f (σ ) ∧ σ ′) > d ′(ρ ′, σ ′) − ε > u. This implies that f (σ ) ∈ T ′k and π ( j) = k, whichcompletes the proof of (106).

Now recall that G = F · φ Γ and that φ(y) = 0, if y ∈ [0, h0]. Thus, (106) implies thatΨu(T ) =

∑j∈JεG(T j ) and Ψu(T ′) =

∑j∈JεG(T ′π ( j)). Recall that G is C-Lipschitz. Then, (103)

and (106) implyΨu(T )−Ψu(T ′) ≤ 76Cε # j ∈ J : Γ (T j ) ≥ h0/2. (107)

To summarise, we have fixed T ∈ Tcpct, u ∈ (0, h) \∆(T ) and we have proved that (107) holdstrue for all sufficiently small ε ∈ (0,∞) and for all T ′ ∈ Tcpct such that δcpct(Rh(T ), Rh(T ′)) <ε/4, Since Rh is δcpct-continuous, this entails (Claim 2) and the proof of Lemma 2.4 (i) iscomplete.

Proof of Lemma 2.4(ii). Since (T, δ) is a Polish space, the Borel Isomorphism Theoremimplies that there exists a one-to-one Borel-measurable function φ : T → R such that itsinverse φ−1

: R → T is also Borel-measurable. Let M =∑

1≤k≤nδTkbe in M (T), where

φ(T1) ≤ · · · ≤ φ(Tn). Then, for all k ∈ N, we define Λk(M) as follows:

Λk(M) = Υ , if k = 0 or if k > n and Λk(M) = Tk , if 1 ≤ k ≤ n.

We next set M f (T) = M ∈ M (T) : ⟨M⟩ < ∞ that is clearly an element of the sigma-fieldGM (T).

Lemma A.2. For all k ∈ N, Λk :M f (T)→ T is measurable.

Proof. We only need to prove that φ Λk is measurable. We set Ax,k := M ∈ M f (T) :φ(Λk(M)) ≤ x for all x ∈ R. Now, we observe that Ax,0 = ∅ if φ(Υ ) > x , that Ax,0 =M f (T),



if φ(Υ ) ≤ x , and that for all k ≥ 1,

Ax,k = (φ(Υ ) ≤ x ∩ ⟨M⟩ < k) ∪

M ∈M f (T) : ⟨M , 1(−∞,x] φ ⟩ ≥ k∈ GM (T) ,

which implies the desired result

Recall from (12) the definition of Paste . It is easy to check that (T , T ′) ∈ T2↦→ T ⊛ T ′ ∈ T

is continuous. Thus, this implies that

M ∈M f (T) ↦−→ Paste (M) = ⊛k∈NΛk(M) is measurable. (108)

Let h ∈ (0,∞). For all M =∑

i∈I δTi, we set Ξh(M) =

∑i∈I δRh (Ti ). Clearly ⟨Ξh(M)⟩ <∞ and

for all measurable F : T→ [0,∞), ⟨Ξh(M), F⟩ = ⟨M, FRh⟩. This implies that Ξh :M (T)→M f (T) is measurable. This result combined with (108) implies that Paste Ξh is measurable.Now, observe that for all h ∈ (0,∞) and all M ∈ M (T), δ (Paste (M),Paste (Ξh(M))) ≤ h,which implies Lemma 2.4 (ii).


By (17), Lemmas 2.3 and 2.4(i), we only need to prove that D is measurable. Let (T, d, ρ) bea CLCR real tree. We first claim the following.

limh→0

D(Rh(T )) = D(T ). (109)

Recall that D(T ) = ∞ iff T has no leaf and no branch point, namely iff T is either a point treeor a finite number of half-lines pasted at their finite endpoint. In these cases, (109) obviouslyholds true. Let us assume that D(T ) is finite. First note that if σ ∈ Br(T ), then σ ∈ Br(Rh(T ))for all sufficiently small h ∈ (0,∞). Let σ ∈ Lf(T ) be such that [[ρ, σ ]] ∩ Br(T ) = ∅. Thenfor all h ∈ (0, d(ρ, σ )), there exists σ ′ ∈ [[ρ, σ ]] such that d(σ ′, σ ) = h. Thus, σ ′ ∈ Lf(Rh(T ))and d(ρ, σ ) = d(ρ, σ ′)+ h. Thus, for all σ ∈ Br(T ) ∪ Lf(T ), d(ρ, σ ) ≥ lim suph→0 D(Rh(T )),which implies that D(T ) ≥ lim suph→0 D(Rh(T )).

Conversely, observe that Br(Rh(T )) ⊆ Br(T ). Next, if σ ′ ∈ Lf(Rh(T )), then there existsσ ∈ Lf(T ) such that σ ′ ∈ [[ρ, σ ]] and d(ρ, σ ′) = d(ρ, σ ) − h. Therefore, for all σ ′ ∈Br(Rh(T )) ∪ Lf(Rh(T )), d(ρ, σ ′) ≥ D(T ) − h. Thus, lim infh→0 D(Rh(T )) ≥ D(T ), whichcompletes the proof of (109).

For all h ∈ (0,∞), we next set Jh(T ) = infa ∈ [0,∞) : Z (h)0 (T ) = Z (h)

a (T ), with theconvention that inf ∅ = ∞. By Lemma 2.4(i), Z (h)

a is measurable and since a ↦→ Z (h)a is caglad,

the function Jh : T → [0,∞] is measurable. Now observe that Jh(T ) ≥ D(Rh(T )) > 0. IfJh(T ) > D(Rh(T )) then, the lowest leaves of Rh(T ) are at the same distance from the rootas the lowest branch points and there exists ε ∈ (0,∞), such that for all h′ ∈ (h, h + ε),D(Rh′ (T )) = Jh′ (T ) = D(Rh(T )) − h′ + h. This implies that lim infh∈Q∩(0,∞)→0 Jh(T ) =lim infh∈Q∩(0,∞)→0 D(Rh(T )), which implies the measurability of D by (109).


Recall that for all n ∈ N, ϑn : T → T and Dn : T → [0,∞] are defined as ϑn+1 = ϑ ϑnand Dn = D ϑn . We set A = T ∈ T :

∑n∈NDn(T ) = ∞, which is a Borel set of T by

Lemma 2.5.Let (T, d, ρ) be a CLCR real tree. Suppose that T ∈ T∗. If T has no leaf and no branch point,

then Dn(T ) = ∞, for all n ∈ N, and it belongs to A. Next assume that there is σ ∈ Br(T )∪Lf(T ).



Then, there exists n0 ∈ N such that∑

0≤k≤n0Dk(T ) = d(ρ, σ ). If d(ρ, σ ) = Γ (T ), then

for all n > n0, Dn(T ) = ∞. Let us assume that Γ (T ) = ∞: for all r ∈ (0,∞), setn(r ) = #(BT (ρ, r )∩(Br(T )∪Lf(T ))), which is finite since T ∈ T∗; the previous arguments implythat

∑0≤k≤n(r ) Dk(T ) > r , which implies that

∑n∈NDn(T ) = ∞. This proves that T∗ ⊆ A.

Conversely, assume that T ∈ A. If D(T ) = 0, then ϑn T = T and Dn(T ) = 0, for all n ∈ N,which contradicts the assumption. If D(T ) = ∞, then T ∈ T∗ (and recall that k(T ) = 0, byconvention). Let us assume that D(T ) ∈ (0,∞), then Blw (T, D(T )) is equivalent to a finitenumber of copies of the interval [0, D(T )] pasted at 0. If k(T ) = ∞, ϑ T has infinitely manytrees pasted at its root and the local compactness implies that there are leaves arbitrarily close toits root, which implies that D1(T ) = 0; therefore Dn(T ) = 0, for all n ≥ 1, which contradicts theassumption. Thus, if D(T ) ∈ (0,∞), then k(T ) < ∞. These arguments and a simple recursionimply first that for all n ∈ N, ϑn T ∈ A and k(ϑn T ) < ∞, and that for all n ∈ N such thatRn := D0(T )+ · · · + Dn(T ) <∞, we get

d(ρ, σ ) ; σ ∈ Br(T ) ∪ Lf(T ) : d(ρ, σ ) ≤ Rn = R0 < R1 < · · · < Rn ,

Thus, if Rn <∞, we get

n(ρ, T )+∑

n(σ, T ) ≤ n(ρ, T )+ (1+ k(T ))+ (1+ k(ϑ1T ))+ · · · + (1+ k(ϑn T ))< ∞ ,

where the first sum is over the branch points σ ∈ Br(T ) such that d(ρ, σ ) ≤ Rn . This easilyentails that (T, d, ρ) satisfies (5) in the definition of discrete real trees.

Appendix B. Proof of the measurability of hereditary reduction

To prove the measurability of hereditary reduction, and also for the lemmas of Section 2.3about GW-real trees, it is useful to be able to push forward distributions between a space ofdiscrete combinatorial trees and the space T∗ of discrete real trees.

B.1. Discrete trees with marks

The discrete combinatorial trees that we consider are rooted, ordered and locally finite. Weuse Ulam’s coding (see Neveu [35]) that allows to view such trees as subsets of the set of finiteinteger words

U =⋃n≥0

(N∗)n ,

where N∗ is the set of positive integers. Here (N∗)0 stands for ∅, where ∅ is the empty word.Before recalling the formal definition of discrete trees in this context, let us set some notation:the concatenation of the two words u = (a1, . . . , am) and v = (b1, . . . , bn) in U is denoted byw = u ∗ v = (a1, . . . , am, b1, . . . , bn). Note that ∅ ∗ u = u = u ∗ ∅. A single-symbol wordshall be denoted by ( j), where j ∈ N∗. The length of u ∈ (N∗)n is denoted by |u| = n, with theconvention |∅| = 0.

For all u ∈ U \ ∅, there exists v ∈ U such that u = v ∗ ( j) for a certain j ∈ N∗. Note that|v| = |u| − 1. We then call v the parent of u and we denote it by←−u . We can view U as a graphwhose set of vertices is U and whose set of edges is ←−u , u ; u ∈ U \ ∅, then we denoteby [[u, v]] the shortest path (with respect to the graph distance) between u and v. We also set]]u, v]] := [[u, v]] \ u and we define similarly [[u, v[[ and ]]u, v[[. For u, v ∈ U, the last commonancestor of u and v is denoted by u ∧ v: we recall that [[∅, v ∧ u]] = [[∅, u]] ∩ [[∅, v]].



Definition B.1. A non-empty subset t ⊂ U is called a tree iff it satisfies the following conditionsfor all u ∈ t.

(a) If u ∈ t is different from ∅, then←−u ∈ t.(b) There exists ku(t) ∈ N, such that v ∈ t : ←−v = u = u ∗ (1), . . . , u ∗ ( ku(t) ) if ku(t) ≥ 1

and v ∈ t : ←−v = u = ∅ if ku(t) = 0.

Note that (a) entails ∅ ∈ t. We view ∅ as the progenitor of the population whose family treeis t. Then, ku(t) stands for the number of children of u ∈ t. We denote by Tdiscr the set of allordered rooted discrete trees.

We view [0,∞] as the compactification of [0,∞) and we denote by ∆ a metric that generates thistopology. We call T = (t; x) a marked tree if t ∈ Tdiscr and if x = (xu, u ∈ t), with xu ∈ [0,∞],for all u ∈ t. We then denote by Tdiscr

[0,∞] :=⨆

t∈Tdiscr(t × [0,∞]t) the set of marked trees. We

equip Tdiscr[0,∞] with the σ -algebra G[0,∞] generated by the subsets

Au,y := (t; x) ∈ Tdiscr[0,∞] : u ∈ t , xu > y , u ∈ U , y ∈ [0,∞]. (110)

Connection with real trees. A discrete tree with finite marks clearly corresponds to a real tree.For technical reason, we associate a real tree to [0,∞]-marked discrete trees with an obviousrestriction due to possibly infinite lifetime marks. More precisely, let T = (t; x) ∈ Tdiscr

[0,∞]. For allu ∈ t, we introduce the following notation:

ζu =∑

v∈[[∅,u]]

xv , u ∈ t. (111)

We can think of ζu as the death-time of u and of ζ←−u as the birth-time of u, with the conventionζ←−∅ = 0. For an obvious reason, we have to assume the following:

∀u ∈ t , ∀v ∈ [[∅, u [[ , ζv <∞ . (112)

We associate with T a rooted real tree denoted by TREE (T) = (T, d, ρ) as follows. We first set

ρ = (∅, 0) and T = ρ ∪ (u, s) ; s ∈ (0, xu] ∩ (0,∞), u ∈ t .

We then define a distance d on T × T as follows: for all σ = (u, s) ∈ T \ ρ, we setd(ρ, σ ) = s +

∑v∈[[∅,u[[xv , which is finite by (112). Let σ ′ = (u′, s ′) ∈ T \ ρ. We then

set

d(σ, σ ′) =

⎧⎨⎩ d(ρ, σ )+ d(ρ, σ ′)− 2∑

v∈[[∅,u∧u′]]

xv , if u ∧ u′ ∈ u, u′,

|d(ρ, σ )− d(ρ, σ ′)| , otherwise.

It is easy to check that TREE (T) := (T, d, ρ) is a rooted real tree. However, note that T mayneither be a discrete real tree nor locally compact. We then introduce

Tdiscr∗=T = (t; x) ∈ Tdiscr

[0,∞] : T satisfies (112) and

∀a ∈ [0,∞) , #u ∈ t : ζu ≤ a <∞. (113)

It is easy to check that if T ∈ Tdiscr∗

, then TREE (T) is a discrete real tree as in Definition 2.1.Next observe that Tdiscr

∗belongs to the sigma-field G[0,∞]. Then, for all T ∈ Tdiscr

∗, we denote by

TREE (T) the pointed isometry class of the discrete real tree TREE (T).

Lemma B.2. TREE : Tdiscr∗−→ T∗ is measurable.



Proof. Let t ∈ Tdiscr be finite and set Ut =x ∈ [0,∞]t

: (t; x) ∈ Tdiscr∗

, which is an open

subset of [0,∞]t equipped with the product topology. First note that x ∈ Ut ↦→ TREE (t, x) isδ-continuous. For any n ∈ N, set En =

⨆(t × [0,∞]t), where the disjoint union is taken over

the set of discrete trees t such that |u| ≤ n for all u ∈ t. Clearly, En is a Polish space when it isequipped with the distance dn that is defined for any T = (t; x), T′ = (t′; x′) ∈ En by

dn(T,T′) = 1 if t = t′ and

dn(T,T′) =∑

0≤m<#t

2−m−1(1 ∧∆(xum , x ′um)) if t = t′,

where u0 = ∅ < u1 < · · · < u#t−1 is the sequence of vertices of t listed in the lexicographicalorder. Next, for all T = (t; x) ∈ Tdiscr

[0,∞] and all n ∈ N, we set t|n = u ∈ t : |u| ≤ nand T|n = (t|n; x|n = (xu, u ∈ t|n)) ∈ En . We then define a metric d on Tdiscr

[0,∞] by settingd(T,T′) =

∑n≥02−ndn(T|n,T′

|n). By standard arguments, (Tdiscr[0,∞], d) is a Polish space. The

previous arguments entail that for any fixed n ∈ N, T ∈ Tdiscr∗↦→ TREE (T|n) is δ-continuous.

Moreover, for any fixed T ∈ Tdiscr∗

, we easily get δ( TREE (T), TREE (T|n)) −→ 0 as n → ∞.This implies that TREE : Tdiscr

∗−→ T∗ is measurable with respect to the d-Borel sigma-field on

Tdiscr[0,∞], which turns out to be G[0,∞].

In the following lemma we prove that TREE has a measurable section. This result is used inthe proof of Lemma 2.15.

Lemma B.3. There is S : T∗→ Tdiscr∗

measurable such that TREE (S(T )) = T , for all T ∈ T∗.

Proof. Recall Lemma A.2 and its notation Λk . First note that for all T ∈ T∗, ⟨M0(T )⟩ =Z+0 (T ) < ∞. Lemmas A.2 and 2.4(i) allow to define for all k ∈ N∗ a measurable functionΦk : T∗→ T∗ by setting Φk(T ) = Λk(M0(T )). Then, for all words u ∈ U we recursively defineφu : T∗→ T∗ by setting for all k ∈ N∗, φ(k)(T ) = Φk(T ), and φu∗(k) = Λk(M0(ϑ(φu(T )))).

If Z+0 (T ) = 0, then T = Υ and we set S(T ) = (∅, 0) that is the progenitor with zerolifetime and no children. Let us assume that n := Z+0 (T ) ≥ 1. Then, for all k ∈ 1, . . . , n,we set tk = u ∈ U : φ(k)∗u(T ) = Υ and for all u ∈ tk , we set xu(k) = D(φ(k)∗u(T )). It iseasy to check that Tk := (tk; x(k)) ∈ Tdiscr

∗. Then, we set S(T ) = ((k) ∗ u, xu(k)); u ∈ tk, k ∈

1, . . . , n ∈ Tdiscr∗

and we easily check that TREE (S(T )) = T . Also, S is clearly measurablesince φu and D are measurable.

B.2. Proof of Lemma 2.8

Let A ⊆ T be hereditary and let h ∈ (0,∞). We set Ah = A ∩ Γ ≥ h. Note that Ah

is hereditary. Let φ stand for a Borel isomorphism from T onto R. Namely, φ : T → R isone-to-one and φ as well as φ−1

: R→ T are Borel-measurable. For all k ∈ N∗, let Lk : T→ Tbe defined as follows. For all T ∈ T, set Mh(T ) =

∑i∈I δTi

and n = ⟨Mh(T ), 1Ah ⟩; then, forall k > n, Lk(T ) = Υ and if n ≥ 1,∑

i∈I

1Ah (Ti )δTi=

∑1≤k≤n

δLk (T ) with φ(L1(T )) ≤ · · · ≤ φ(Ln(T )) .

Lk(Υ ) = Υ . We argue as in Lemma A.2 to prove that Lk is measurable.Then for every word u ∈ U, we define a measurable function Lu : T→ T such that L∅ is the

identity map on T, L (k) = Lk , for all k ∈ N∗ and Lv Lu = Lu∗v , for all u, v ∈ U.



We next fix a CLCR real tree (T, d, ρ) and we define a [0,∞)-marked discrete tree (t; x) ∈Tdiscr

[0,∞) by setting

t = ∅ ∪u ∈ U \ ∅ : Lu(T ) = Υ

and ∀u ∈ t , xu = h .

We then set (T ′, d ′, ρ ′) = TREE (t; x). Recall that each edge of T ′ has length h and correspondsto one vertex in t. Recall that formally, ρ ′ = (∅, 0) and T ′ = ρ ′ ∪ (u, s); s ∈ (0, h], u ∈ t.We denote by Φh(T ) the pointed isometry class of T ′. Recall from Lemma B.2 that TREE ismeasurable. Since the functions Lu , u ∈ U, are measurable, Φh : T→ T is then measurable.

We now prove that T ′ and RA(T ) are close with respect to δ. To that end, for all ℓ ∈ N, we set

Sℓ = ρ ∪σ ∈ T : d(ρ, σ ) = ℓh and ⟨M0(θσT ), 1Ah ⟩ ≥ 1

and S :=

⋃ℓ∈N

Sℓ .

It is easy to see that S is a (2h)-net of RA(T ). Indeed, let σ ∈ RA(T ). There exists ℓ ∈ N suchthat ℓh ≤ d(ρ, σ ) < (ℓ + 1)h. If ℓ = 0, then d(ρ, σ ) ≤ h, which entails d(σ, S) ≤ h. Assumethat ℓ ≥ 1 and let σ ′ ∈ [[ρ, σ ]] be such that d(ρ, σ ′) = (ℓ−1)h. Then, d(σ, σ ′) ∈ [h, 2h). Denoteby T

∗the connected component of θσ ′T \ σ ′ that contains σ and set T∗ = T

∗∪ σ ′. Note that

T∗ is an atom of M0(θσ ′T ) and that Γ (T∗) ≥ h. Since σ ∈ T ∗∩ RA(T ), there exists σ ′′ ∈ T

∗

such that θσ ′′T ∈ A. Since θσ ′′T = θσ ′′T∗, we then get T∗ ∈ A. This implies that σ ′ ∈ Sℓ−1.Thus, d(σ, S) ≤ 2h, which proves that S is a (2h)-net of RA(T ).

From the definition of the functions Lu and of the tree t, we easily check that there is a functionȷ : t→ S that satisfies the following property: ȷ is surjective, ȷ (∅) = ρ, and for all u ∈ t \ ∅,Lu(T ) is an atom of M0(θȷ (u)T ) and d(ρ, ȷ (u)) = |u|h. We now define f : T ′ → S by settingf ((u, s)) = ȷ (u), for all (u, s) ∈ T ′. We easily see that

∀ (u, s), (u′, s ′) ∈ T ′ ,d ′((u, s), (u′, s))− d(ȷ (u), ȷ (u′))

≤ 2h ,

which implies that the distortion of f is less than 2h. Since f (T ′) = S is a (2h)-net of RA(T ), fis a pointed (2h)-isometry and Lemma A.1 implies that δ(T ′, RA(T )) ≤ 8h. This proves that forall h ∈ (0,∞), there exists a measurable function Φh : T→ T, such that

∀ T ∈ T , δ(Φh(T ), RA(T )

)≤ 8h ,

which completes the proof of Lemma 2.8.

Appendix C. Proofs of the preliminary results on GW-trees

C.1. Proof of Lemma 2.15

Lemma 2.15 looks obvious, but it is not. Since it is the point of entrance of GW-laws into thespace T, we proceed with care in several steps.

Step 1: existence. Here we use a construction in Tdiscr[0,∞). For all u ∈ U, we define the u-shift

θu , by setting for all w = u ∗ v, θuw = v. For every subset A ⊆ U, we also define θu A as the(possibly empty) set of words v ∈ U such that u ∗ v ∈ A. For all u ∈ t, we set θuT = (θut; θux),where θux = (xu∗v, v ∈ θut), and we slightly abuse notation by writing u ∈ T instead of u ∈ tand ku(T) instead of ku(t).

Let us fix an offspring distribution ξ and c ∈ (0,∞). Let (Ω ,G,P) be a probability space onwhich is defined a family (N (u), xu)u∈U of i.i.d. N × [0,∞)-valued r.v. with law ξ ⊗ ce−cx dx .We then set

τ = ∅ ∪⋃n≥1

u = (a1, . . . , an) ∈ (N∗)n

: ∀k ∈ 1, . . . , n, 1 ≤ ak ≤ N(a1,...,ak−1),



with the convention that (a1, . . . , ak−1) = ∅ if k = 1. We set x = (xu, u ∈ τ ) andT = (τ ; x). Clearly, T ∈ Tdiscr

[0,∞]. Recall from (110) the definition of the elementary sets Au,y .We immediately see that T ∈ Au,y ∈ G, which entails that T : Ω → Tdiscr

[0,∞] is (G,G[0,∞])-measurable. Moreover, we easily check that T satisfies the following two properties.

(a) The law of k∅(τ ) is ξ , the law of x∅ is η(dx) = ce−cx dx , and k∅(τ ) and x∅ areindependent.

(b) For all k ∈ N∗ such that ξ (k) > 0, the subtrees ( θ( j)(T ); 1 ≤ j ≤ k ) underP( · | k∅(τ ) = k) are i.i.d. copies of T under P, and they are independent of x∅ .

Furthermore, if T ′ also satisfies (a) and (b), we check that for all n ≥ 1, all u1, . . . , un ∈ U, ally1, . . . , yn ∈ [0,∞], P(T ∈ Au1,y1∩· · ·∩Aun ,yn ) = P(T ′ ∈ Au1,y1∩· · ·∩Aun ,yn ) and a monotoneclass argument entails that T and T ′ have the same law, which we call the GW(ξ, c)-distributionon Tdiscr

[0,∞] and which is therefore characterised by (a) and (b). We refer to Neveu [35], for moredetails.

For any a ∈ [0,∞), we then set Z+a (T ) = #u ∈ τ : ζ←−u ≤ a < ζu ∈ N ∪ ∞, that is thenumber of individuals that are alive at time a. Properties (a) and (b) imply that up to a possibleexplosion in finite time, the process a ↦→ Z+a (T ) is a continuous-time N-valued Markov chainwhose matrix-generator (qi, j )i, j∈N is given by qi,i = −ci , qi, j = 0 if j < i − 1, qi,i−1 = ciξ (0)and qi, j = ciξ ( j − i + 1) if j > i . Standard analytical computations imply that a.s. explosiondoes not occur iff ξ is conservative as defined in (27): see [5, Section III.2, Theorem 1] formore details. Then, if ξ is conservative, a ∈ [0,∞) ↦→ Z+a (T ) is N-valued and cadlag a.s.Thus,

ξ conservative H⇒ P-a.s. T ∈ Tdiscr∗

. (114)

Let us furthermore assume that ξ is proper. Then, the law on T of TREE (T ) satisfies (a) inDefinition 2.14. Note that this law is concentrated on T∗. This proves that for every properconservative offspring distribution ξ and every c ∈ (0,∞), there exists at least one probabilitymeasure on T∗ that satisfies (28) in Definition 2.14.

Step 2. Let Q be as in Definition 2.14. We claim that Q(T∗) = 1.

Proof. On the auxiliary probability space (Ω ,G,P), we consider an N-valued Markov processZ = (Z t )t∈[0,∞) with initial state Z0 = 1 and with matrix-generator (qi, j )i, j∈N as defined above.We then set J0 = inft > 0 : Z t = Z0 and for all n ∈ N, Jn+1 = inft > Jn : Z t = Z Jn , withthe convention that inf ∅ = ∞. Then, (Jn)n∈N are the jump times of Z and if Z is absorbed attime Jn , Jp = ∞, for all p > n. Since ξ is assumed to be conservative, Jn →∞ a.s. as n→∞.

Let Q be a law on T as in Definition 2.14. It is easy to prove recursively that D0 + · · · + Dn

under Q has the same law as Jn under P. This implies that Q-a.s.∑

n≥0 Dn = ∞, which impliesthe claim by Lemma 2.6.

Step 3. Let Q be as in Definition 2.14 and suppose that Q satisfies (28) with (ξ, c) and (ξ ′, c′).Then (ξ, c) = (ξ ′, c′) follows straight from (28).

Step 4. Conversely: let ξ be proper and conservative, let c ∈ (0,∞). Suppose that Q and Q′

satisfy (28) in Definition 2.14. Then, we claim that Q = Q′.



Proof. Recall the function S from Lemma B.3. In the definition of S, the vertices have beenordered in a way that causes a lack of exchangeability. This is why we introduce a shuffling kernelas follows. Let T ∈ Tdiscr

∗. Denote by K (T, dT′) the law of the discrete marked tree obtained by

permuting independently and uniformly the siblings (with their corresponding lifetime marks).It is easy to check that K is a measurable kernel. Then by Lemma B.3, it is easy to check thatthe two laws P :=

∫Q(dT )K (S(T ), dT) and P ′ :=

∫Q′(dT )K (S(T ), dT) satisfy (a) and (b) of

the definition of a discrete Galton–Watson distribution. As already mentioned there is a uniqueGW(ξ, c) law on Tdiscr

[0,∞]. Therefore, P = P ′. But now observe that Q is the law of TREE underP and that Q′ is the law of TREE under P ′. Thus, Q = Q′, which entails the desired result. Thisfinishes the proof of Lemma 2.15.


Basic computations. Before proving Lemma 2.16, let us prove some basic facts. Let us fix aproper conservative offspring distribution ξ and c ∈ (0,∞). Recall that Tdiscr stands for the setof (ordered rooted) discrete trees with no marks as in Definition B.1. We then set

Tdiscrf = t ∈ Tdiscr

: #t <∞. (115)

Let t ∈ Tdiscrf . Recall that for all u ∈ t, ku(t) stands for the number of children of u. We denote

by Lf(t) = u ∈ t : ku(t) = 0 the set of leaves of t. For each subset S ⊆ Lf(t), we define thefollowing weight

wξ (t, S) =∏

u∈t\S

ξ (ku(t)) . (116)

Let x = (xu)u∈t ∈ [0,∞)t, so that T = (t; x) is a [0,∞)-marked tree. Recall from (111) notationζu and ζ←

ufor resp. the death time and the birth time of u ∈ t. We also set L(x) =

∑u∈txu . For

all a ∈ (0,∞), we define

Dt,S,a =

x = (xu)u∈t ∈ [0,∞)t: ζu < a if u ∈ t \ S and xu = a − ζ←

uif u ∈ S

.

We also introduce the following finite measure on [0,∞)t:

Mct,S,a(dx) := 1Dt,S,a (x)c#t−#Se−cL(x)

∏u∈S

δa−ζ←u

(dxu)∏

u∈t\S

dxu , (117)

where δb(dy) stands for the Dirac mass at b ∈ [0,∞).

Let T = (τ ; x) have the GW(ξ, c)-distribution on Tdiscr[0,∞], as defined in Appendix C.1 (and

recall that T satisfies (a) and (b)). We then set S = u ∈ τ : ζ←u< a ≤ ζu. We list S in the

lexicographical order and write S = u1 < · · · < u#S. The forest of discrete trees above levela is then given by Fa

=(θuℓ,aT

)1≤ℓ≤#S, with the convention that Fa is a cemetery point ∂ if

S = ∅. The tree below level a is then given by T a = (τ a; xa), where τ a = u ∈ τ : ζ←u< a,

and where xau = xu , if u ∈ τ a \ S and xa

u = a − ζ←u

if u ∈ S. We next denote by Πn(dF)the law of a finite sequence (namely, a forest) of n independent GW(ξ, c)-discrete trees (withthe convention that Π0 is the Dirac mass on the cemetery point ∂). For all measurable functionsG1 : ∂ ⊔

⨆n≥1(Tdiscr

[0,∞])n→ [0,∞) and G2 : Tdiscr

[0,∞] → [0,∞), we easily get for all t ∈ Tdiscrf ,

for all S ⊆ Lf(t) and for all a ∈ (0,∞),

E[G1(Fa)G2(T a)1τa=t;S=S

]= wξ (t, S)Π#S[G1]

∫Dt,S,a

Mct,S,a(dx) G2(t; x) . (118)



We also set

paξ,c(t) := E[1τa=t] =

∑S⊆Lf(t)

wξ (t, S)⟨Mct,S,a⟩ , (119)

where ⟨Mct,S,a⟩ stands for the (finite) mass of the measure Mc

t,S,a . Recall from (114) that since ξis proper and conservative, T ∈ Tdiscr

∗a.s. and thus,∑

t∈Tdiscrf

paξ,c(t) = 1. (120)

Now set T = TREE (T ) that is a GW(ξ, c)-real tree, whose law on T is Qξ,c. Note thatAbv (a, T ) = TREE (Fa), that Blw (a, T ) = TREE (T a), and that Z+a (T ) = #S. Thus, (118)implies that for all measurable functions F,G : T→ R+, for all n ∈ N and all a ∈ (0,∞),

Qξ,c

[F (Abv (a, ·)) G (Blw (a, ·)) 1

Z+a =n

]= Qξ,c

[Q⊛nξ,c[F] G (Blw (a, ·)) 1

Z+a =n

]. (121)

Note that for any a ∈ [0,∞), there is no u ∈ τ , such that a = ζu , a.s. Thus, b ↦→(Z+b (T );Blw (b, TREE (T ))) is continuous at time a, a.s. Namely,

∀a ∈ [0,∞), Qξ,c-a.s. b ↦→(Z+b ,Blw (b, · )

)is continuous at time a. (122)

This implies in particular that (121) holds true with a = 0.

Proof of Lemma 2.16. Note that (121) and (122) prove that (iii) H⇒ (ii). The implication(ii) H⇒ (i) is obvious. It only remains to prove (i) H⇒ (iii). So, we assume that Q is a probabilitymeasure on T that satisfies (i), and that is such that Q(0 < D < ∞) > 0 and such that Z+ isQ-a.s. cadlag.

We first prove that Q(Z+0 = 1) = 1. Indeed, by (i) with a = 0, Q(Z+0 = n) = Q(Z+0 =n)Q⊛n(Z+0 = n), for all n ≥ 1. Suppose that there exists n ≥ 2 such that Q(Z+0 = n) > 0,then Q⊛n(Z+0 = n) = 1, which implies that Q⊛n(Z+0 = k) = 0 if k = n. But Q⊛n(Z+0 =n2) ≥ Q(Z+0 = n)n > 0, which is impossible. Thus, Q(Z+0 = 0) + Q(Z+0 = 1) = 1. SinceQ(D = ∞) < 1, Q(Z+0 = 0) < 1, which implies that Q(Z+0 = 1) > 0. Now observe that (i)entails Q(Z+0 = 1) = Q(Z+0 = 1)2, so we get Q(Z+0 = 1) = 1.

This implies that for all a ∈ [0, D), Z+a = 1 = Z+D . Namely, D = infa ∈ [0,∞) : Z+a =Z+0 . We next fix b ∈ (0,∞), p ∈ N, and G : T → R, bounded and measurable. By (i) witha = 2−pb, we get

Q[G(Abv (b, ·))1⋂

1≤k≤2p Z+k2−pb

=1

]= Q

(Z+2−pb = 1

)Q[G(Abv (b − b2−p, ·))1⋂

1≤k≤2p−1Z+

k2−pb=1

].

An easy inductive argument implies that

Q[G(Abv (b, ·))1⋂

1≤k≤2p Z+k2−pb

=1

]= Q[G]Q

(Z+b2−p = 1

)2p

= Q[G]Q

⎛⎝ ⋂1≤k≤2p

Z+k2−pb = 1

⎞⎠ .



Since a ↦→ Z+a is Q-a.s. cadlag, limp→∞1⋂1≤k≤2p Z+

k2−pb=1 = 1D>b and a simple argument and

the assumption Q(0 < D <∞) > 0 imply that there is c ∈ (0,∞) such that Q(D > b) = e−cb.Namely, D under Q has an exponential law with mean 1/c and we get

Q[G(Abv (b, ·))1D>b

]= Q[G] Q(D > b). (123)

Recall that ⌈·⌉ stands for the ceiling function and set Dp = 2−p⌈2p D⌉ that decreases to D as

p →∞. Fix n ∈ N \ 1, F : T→ R and f : [0,∞)→ R, bounded and continuous. We thenset

Ak,p = Q[F(Abv (k2−p, ·)) f (k2−p)1

Dp=2−pk;Z+Dp=n

]and

Ap =∑k∈N

Ak,p = Q[F(Abv (Dp, ·)) f (Dp)1

Z+Dp=n

].

Then, limp→∞Ap = Q[F(ϑ) f (D)1k=n

], by Lemma 2.3 and since Z+ is assumed to be Q-

a.s. cadlag. We next apply (123) with b = (k − 1)2−p, and then (i) with a = 2−p to get

Ak,p = Q⊛n[F]Q(Z+2−p = n

)e−c(k−1)2−p

f (k2−p).

By taking F and f equal to 1, we get Q(Dp = 2−pk; Z+Dp= n) = Q

(Z+2−p = n

)e−c(k−1)2−p

.Summing over k ≥ 1 entails Q

(Z+2−p = n

)= (1− e−c2−p

)Q(Z+Dp= n). Thus

Ak,p = Q⊛n[F]Q(Z+Dp= n

)(1− e−c2−p

)e−c(k−1)2−pf (k2−p)

= Q⊛n[F]Q(Z+Dp= n

)Q(Dp = k2−p) f (k2−p).

Summing over k ≥ 1 entails Ap = Q⊛n[F]Q(Z+Dp= n)Q[ f (Dp)], which implies

Q[F(ϑ) f (D)1k=n] = Q⊛n[F]ξ (n)∫∞

0 f (x)ce−cx dx , where ξ (n) = Q(Z+D = n), for all n ∈ N.Namely, Q is the law of a GW(ξ, c)-real tree.


The statement for GW-forests is easily derived from the analogous result for single GW-realtrees. We only need to prove that for all a ≥ b ≥ 0, all p ∈ N and all bounded measurablefunctions F : T→ R:

limn→∞

Qξn ,cn [F (Blw (a, · ))] = Qξ∞,c∞ [F (Blw (a, · ))] . (124)

Recall (119) and (120). Thus, for all n ∈ N ∪ ∞, we have, in the notation of (115), (116) and(117), that

Bn := Qξn ,cn [F(Blw (a, · ))]

=

∑t∈Tdiscr

f

∑S⊆Lf(t)

wξn (t, S)∫

Dt,S,a

Mcnt,S,a(dx) F

(TREE (t; x)

).

Observe that for all t ∈ Tdiscrf and all S ⊆ Lf(t), limn→∞wξn (t, S) = wξ∞ (t, S),

limn→∞

∫Mcn

t,S,a(dx) F(

TREE (t; x))=

∫Mc∞

t,S,a(dx) F(

TREE (t; x))

(125)



and limn→∞ paξn ,cn

(t) = paξ∞,c∞ (t). Thus, for all finite subsets A ⊂ Tdiscr

f , we get

lim supn→∞

|Bn − B∞| ≤ ∥F∥ paξ∞,c∞

(Tdiscr

f \ A)+ ∥F∥ lim

n→∞paξn ,cn

(Tdiscr

f \ A)

= 2∥F∥(1− pa

ξ∞,c∞ (A)),

which implies (124) because paξ∞,c∞ is a probability function on the countable set Tdiscr

f .

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